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National Bureau of Standards 
Library, N.W. Bldg 




AUG 8 1962 



^ecltnical ^lote 



100 



REQUIRED SIGNAL-TO-NOISE RATIOS, 

RF SIGNAL POWER, AND BANDWIDTH FOR 

MULTICHANNEL RADIO COMMUNICATIONS SYSTEMS 



E. F. FLORMAN AND J. J. TARY 




U. S. DEPARTMENT OF COMMERCE 
NATIONAL BUREAU OF STANDARDS 



THE NATIONAL BUREAU OF STANDARDS 

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NATIONAL BUREAU OF STANDARDS 

technical ^Kiote 

100 

JANUARY, 1962 

ORIGINAL: JULY, 1960 

REVISED: NOV., 1960; JUNE, 1961; AND NOV. 1961 



REQUIRED SIGNAL-TO-NOISE RATIOS, 

RF SIGNAL POWER, AND BANDWIDTH FOR 

MULTICHANNEL RADIO COMMUNICATIONS SYSTEMS 

E.F. Florman and J.J. Tary 
NBS Boulder Laboratories 



NBS Technical Notes are designed to supplement the Bu- 
reau's regular publications program. They provide a 
means for making available scientific data that are of 
transient or limited interest. Technical Notes may be 
listed or referred to in the open literature. 



For sale by the Superintendent of Documents, U.S. Government Printing Office 
Washington 25, D.C. - Price $1.00 



CONTENTS 

Page 

No. 

1. INTRODUCTION 1 

2. RADIO COMMUNICATION SYSTEM 4 

2. 1 Performance of System 5 
2.2 Design and Testing of Equipment and Systems ' 7 

3. MODULATOR -TRANSMITTER PERFORMANCE 9 

3. 1 Required Radio Frequency Signal-Spectrum 10 

Bandwidth for Radio Communication Systems 

3. 1. 1 Radio Frequency Bandwidth Required for 10 

SSBSC-AM Systems 

3. 1.2 Radio Frequency Bandwidth Required for 10 

FM Systems 

3. 1.2. 1 Required FM Radio Frequency 11 

Bandwidth for a Single Sinsoidal 
Modulating Signal 

3.1.2.2 Required FM Radio Frequency 15 
Bandwidth for Various Types of 
Modulating Signals 

3.1.2.3 Required FM Radio Frequency 16 
Bandwidth for a "White Noise 
Modulating Signal 

3.2 Radio -Frequency Signal Time -Amplitude 23 

Characteristics for Radio Communication Systems 

3.2.1 Radio-Frequency Signal Time -Amplitude 23 
Characteristics for SSBSC-AM Systems 

3.2.1.1 Radio-Frequency Signal Time- 23 

Amplitude Characteristics of 
SSBSC-AM Systems for a Single 
Sinusoidal Modulating Signal 



111 



Page 
No. 

3.2.1.2 Radio -Frequency Signal Time- 24 

Amplitude Characteristics of 
SSBSC-AM Systems for a White- 
Noise Type of Modulating Signal 

3. 2. 2 Radio-Frequency Signal Time -Amplitude 27 
Characteristics for FM Systems 

3. 3 Peak Power Requirements for SSBSC-AM Systems 27 
RADIO RECEIVER -COMBINER PERFORMANCE 29 

4. 1 Radio Frequency Signal Characteristics and 30 

Power Levels in the Receiving System 

4.2 Pre-Detection Noise-Power Levels in the Re- 33 

ceiving System 

4. 3 Post-Detection Noise-Power Levels in tne Re- 41 

ceiving System 

4. 3. 1 Post-Detection Noise Due to Pre-Detec- 43 

tion Amplitude Noise 

4. 3. 2 . Post-Detection Noise Due to Noise at the 43 

Modulator Input 

4.3.3 Equipment -Intermodulation Post-Detection 44 
Noise 

4.3.4 Post-Detection Path-Modulation Noise 47 

4.3.5 Total Post-Detection Noise 49 

4. 3. 6 Effect of Post-Detection Noise on 50 

Receiver Characteristics and Methods of 
Correcting System -Design Curves 

4.4 System Parameters which Affect the Message- 53 

Channel Signal -to-Noise Ratio at the Radio 
Receiver Output 

4.4.1 Dependence of Receiver-Output Signal-to- 54 

Noise Ratio on the Modulating -Signal 

Power Level, P 

m 



IV 



Page 

No. 

4.4.2 Dependence of Receiver -Output 56 

Message-Channel Signal-to-Noise Ratio, 

S /N , on the Message Channel Signal- 

oc oc / 

Power Loading Ratio, P /p , in the 

Modulating Baseband Signal. 

4.5 Performance Characteristics of SSBSC-AM Radio 59 
Receivers 

4. 5. 1 Receiver -Input Radio Frequency Signal 6l 

Power Requirements for SSBSC-AM 
Systems Using Steady Received 
Signals 

4. 5.2 Receiver -Input Radio Frequency Signal 66 

Power Requirement for SSBSC-AM Systems 
Using a Rayleigh-Fading Type of Received 
Signal 

4.6 Performance Characteristics of FM Radio Re- 67 
ceiver s 

4. 6. 1 FM Radio Receiver Characteristic Curves 68 

4. 6. 1. 1 Linear Region of FM Receiver 69 

Characteristic Curves 

4.6.1.2 Threshold Region of FM Receiver 71 
Characteristic Curves 

4. 6. 2 FM Radio Receiver Input -Power and 73 

Bandwidth Requirements for Steady 
Received Signals 

4.6. 3 FM Radio Receiver Input-Power and Band- 77 
width Requirements for a Rayleigh-Fading 
Type of Received Signal 

4.6.4 Pre-Emphasis - - De-Emphasis Techniques 81 

4. 6. 5 Bandwidth Compression in FM Receivers 83 



Page 

No. 

PERFORMANCE CHARACTERISTICS OF MESSAGE- 88 

SIGNAL DECODER UNIT 

RADIO COMMUNICATION SYSTEM DESIGN AND TEST 95 
PROCEDURES 

6. 1 Determination of the Basic System Parameters: 97 

S /N , B , P /p , f , T A and T for 
oc oc c m m m A er 

Specified Average Message Error Rates, Number 

of Available Message Channels, and Specified 

Receiving -System Parameters. 

6.2 Equipment Design Procedures, SSBSC-AM 101 

Systems, for a Rayleigh-Fading Type of Received 
Signal, Dual Diversity, and Maximal -Ratio 
Combining 

6. 3 Equipment Design Procedures, FM Systems, 103 

for a Rayleigh-Fading Type of Received Signal, 
Dual Diversity, and Maximal -Ratio Combining 

6.4 Test Procedures for SSBSC-AM and FM Systems 108 

6. 5 System Performance Estimates for SSBSC-AM 108 

and FM Systems 

REQUIRED TRANSMITTER-OUTPUT POWER 114 

CONCLUSIONS 119 

ACKNOWLEDGEMENTS 121 

REFERENCES 122 

LIST OF SYMBOLS 126 

LIST OF FIGURES 132 



VI 



Page 
No. 



APPENDICES 



A. PRE-DETECTION NOISE-POWER LEVELS IN 

THE RECEIVING SYSTEM 135 

B. DERIVATION OF THE MESSAGE -CHANNEL 
SIGNAL -POWER LOADING RATIO P^/p , IN 

THE MODULATING BASEBAND SIGNAL 163 

C. DERIVATION OF FM RECEIVER PERFORMANCE 
CHARACTERISTICS 167 

D. METHOD OF ESTIMATING THE PERFORMANCE 
CHARACTERISTICS OF RADIO RECEIVERS AND 
MESSAGE -SIGNAL DECODING UNITS, FOR A 
TIME -VARYING SIGNAL 174 

D. 1. Estimated Performance Characteristics of 

Radio Receivers 174 

D. 1. 1 Estimated Performance of Radio 

Receivers with Diversity j 78 

D. 2 Estimated Performance Characteristics 
of the Decoder Unit, for a Time -Varying 
Input Signal-to-Noise Ratio 1?9 



vn 



ABSTRACT 
A method is outlined for determining the relationships between 
the grade of performance, or message error rate, of a radio 
communication system and the system parameters. Results are 
presented in the form of design equations and system-design curves 
The system parameters and variables considered are the following 
definable and measurable factors: signal-to-noise ratios, carrier- 
signal power level, message load, receiver noise figure, etc. 
These factors are used with the design curve scales as normalizing 
factors in order to yield quantitative results. 



vm 



REQUIRED SIGNAL-TO-NOISE RATIOS, RF SIGNAL POWER, 
AND BANDWIDTH FOR MULTICHANNEL 
RADIO COMMUNICATION SYSTEMS 

by 
E. F. Florman and J. J. Tary 

1. INTRODUCTION 

The objective of this work is to obtain the basic relationships be- 
tween the grade of service, or the message error-rate, and the 
various system parameters, for radio communication systems. The 
results are presented in the form of design equations and sets of 
system-design curves, in which the system parameters are used as 
variables and as scale -normalizing factors. The design curves 
are basic and are general in form; however, they may easily be con- 
verted to families of curves which apply directly to specific cases, 
involving specified system parameters. 

This paper outlines a method for obtaining design equations and 
sets of design curves for radio communication systems, based pri- 
marily on the combined performance characteristics of the radio 
receiver, the diversity combiner, and the message-signal decoder 
unit. Results are obtained in terms of the important specified, defin- 
able, and measurable system parameters, such as message error 
rates, modulating -signal power levels, modulating baseband signal 
bandwidth, signal-to-noise ratio at the radio receiver output, radio 
receiver noise figure, order of diversity, number of available mes- 
sage channels, etc. Through the application of these results it is 
possible to determine directly the optimum values for the important 
factors, such as the required radio-frequency signal power and the 
optimum bandwidth. 

-1- 



-2- 

It is assumed that the grade of service for the system is directly- 
associated with the radio receiver output signal-to-noise ratio; there- 
fore, this ratio is used as a primary factor in determining the system 
performance . 

In this work the radio receiver output message-channel signal-to- 
noise ratio, S /N , is determined in terms of: 
oc oc 

(1) the radio- receiver input total RF- signal power and the total 

noise power levels, 

(2) the pre-detection signal-to-noise ratio, 

(3) the receiver noise figure, 

(4) the relative power levels of the modulating signals at the 

radio transmitter, 

(5) the power- spectrum bandwidth of the composite modulating 

signal, 

(6) the power-spectrum bandwidth of the modulated radio-fre- 

quency signal, 

(7) the modulation index, and 

(8) position of the signal in the receiver-output baseband signal 

spectrum. 

The effect of other factors on the receiver output signal-to-noise 
ratio may also be incorporated; these additional factors include: 

(9) effects of bandwidth compression in the radio receiver IF, 
or at the pre-detection point, 

(10) radio frequency path bandwidth capability, or effects of fre- 
quency selective fading on the radio-frequency signal, and 

(11) intermodulation noise generated within the equipment. 

The statistical characteristics of the signal-to-noise ratio at the 
output of the radio receiver depend upon the time-varying character- 
istics of (a) the received radio-frequency signal and (b) the total noise 
power, at the receiver input. The combined radio receiver output 
signal-to-noise characteristics also depend upon the order of diversity, 
type of diversity, type of combining, and the point within the receiver 



-3- 
at which combining takes place, such as pre-detection and/or post- 
detection. For example, the distribution of the receiver-output 
signal-to-noise ratio, for a time-varying received RF signal, may be 
determined by combining the steady- signal characteristic of the radio 
receiver, with the combiner characteristics, for a particular comu- 
lative distribution of the time -varying received radio-frequency sig- 
nal. This procedure is used here to determine the cumulative distri- 
bution of the combined receiver- output signal-to-noise ratio, for 
various orders of diversity; assuming that the received radio-fre- 
quency carrier signal is Rayleigh distributed. Other types of received 
RF signals may also be considered, provided that the statistical char- 
acteristics of these signals are known. 

The cumulative distribution of the short-term message error 
rate is obtained by combining the steady- signal performance char - 
acteristic of the message-signal decoder unit with the cumulative 
distribution of the time-varying signal-to-noise ratio at the receiver 
output, or at the decoder-unit input. From the resultant cumulative 
distribution of the message error-rate we obtain the performance 
characteristics of the decoder unit for a time-varying signal-to-noise 
ratio at the decoder unit input. The above procedures are used for 
non, dual and quadruple diversity. 

It should be noted in the above analysis that the performance of 
the radio receiver is considered in terms of the total RF-signal and 
total noise powers at the receiver input versus the receiver output 
signal-to-noise ratio. The performance of the decoder unit is taken 
to be in terms of the input signal-to-noise ratio versus the output- 
message error rate. The reason for the difference between the se- 
lected input -terminal factors for these equipment units is that the RF- 
signal power at the receiver input is a separable factor, which may 



-4- 
be adjusted independently (within limits) while the signal-to-noise 

ratio at the receiver output or the decoder unit input, must be treated 

as a ratio because of the characteristics of the radio receiver. 

2. RADIO COMMUNICATION SYSTEMS 

From a technical viewpoint, a radio communication link should 
transfer information between two or more points at a specified rate, 
for a given amount of transmitted power, a required RF spectrum 
bandwidth, and with an acceptably small received-information error 
rate. An exchange between these major requirements can be effected; 
however, their mutual interdependence usually cannot be eliminated. 
Decisions involving the choice of the type of communication system to 
be used involve such factors as total initial cost, operational cost, 
time of delivery of equipment, etc. ; most of these factors are obtain- 
able from a technical evaluation of the various types of systems. 

This paper deals with the comparative performance of point-to- 
point radio communication systems employing either SSBSC-AM 
(single sideband suppressed carrier, amplitude modulated) or FM 
(frequency modulated) types of modulation. Frequency division type 
of multiplexing is considered and the number of voice channels ranges 
from 1 to 120, or more. However, the subject is treated on the basis 
of message-error rates, required baseband frequency bandwidth, 
available received radio carrier power levels, signal-to-noise ratios, 
etc. Hence, the results are not restricted to f requency-division 

type of multiplexing nor to broadband systems, and are of general use 
in the field of electronic communications. 

Results are presented in a form which should be useful to the 
system designer who has the responsibility of selecting the optimum 
combination of parameters; the results can also be used as a guide in 
setting up system performance tests. 



-5- 
The method of analyzing and evaluating a radio communication 
system, as outlined below, is to consider the performance character- 
istics of the major sections of the system and then optimize the per- 
formance of each of these sections. By this procedure, it is possible 
to compare directly the overall performance and relative merits of 
different types of equipment selected to perform the same function 
within a communication system. Referring to figure 2-1, the major 
sections comprising a radio communication system are seen to be: 

(A) The encoding units which accept and encode the information 
into the form .of an electronic message -signal, and a multi- 
plexing unit which forms a composite baseband signal from 
the set of encoded information-bearing message signals. 

(B) The modulator -radio transmitter transmitting- antenna sec- 
tion which generates, modulates and then directs a radio- 
wave signal to the radio receiver. This baseband- signal 
modulated radio-wave signal acts as a carrier to convey the 
baseband signal to the receiving end of the communication 
system. 

(C) The radio-wave transmission paths having characteristics 
which are dependent upon such factors as antenna beam width, 
radio-carrier frequency, radio frequency spectrum, path 
length, etc. 

(D) The receiving antenna, radio receiver^and diversity-combiner 
unit, considered as a section which accepts the modulated 
radio-frequency signals and delivers a replica of the trans- 
mitted composite baseband signal. 

(E) The receiving multiplexing unit and decoding unit combined, 
which converts the received baseband signal to information in 
its original form. 

2. 1 Performance of System 

Optimum performance of a radio communication system can be 
achieved only by proper considerations of the design parameters in- 
volved in the functioning of the various sections of the system. Refer- 
ring to figure 2-1, the factors to be considered in this paper are: 



BLOCK DIAGRAM OF BASIC UNITS 
IN A MULTICHANNEL TROPOSPHERIC RADIO COMMUNICATION SYSTEM 



TRANSMITTER TRANSMISSION 
LINE 




RADIO RECEIVERS 




-RADIO WAVE 
TRANSMISSION PATHS 




.MODULATING OR 
BASEBAND SIGNAL 



RADIO RECEIVER 
OUTPUTS 

RECEIVED COMBINED" 
BASEBAND SIGNAL 



MULTIPLEX 

(TRANSMIT) 

UNIT 



MULTIPLEX 

(RECEIVE) 

UNIT 



-CODED MESSAGE- SIGNAL CHANNELS 



ENCODING 
UNITS 



INFORMATION BEARING 
CHANNELS 

(TO BE TRANSMITTED) 




INFORMATON BEARING 
CHANNELS 
(RECEIVED) 



Figure 2-1 



-7- 

(1) Spectrum bandwidth required for the composite baseband 

signal, which contains the information to be transmitted. 

(2) Cumulative amplitude distribution of the composite baseband 
or modulating signal; that is, the modulating- signal power 
levels exceeded for various percentages of the time. 

(3) Required transmitted RF spectrum bandwidth. 

(4) Signal-to-noise ratio in the radio receiver IF circuits at the 
demodulator input, and the statistical characteristics of this 
(received) signal-to-noise ratio. 

(5) Received power level, P , at the radio receiver input and the 

characteristics of this received signal such as the cumulative 
amplitude distributions, fade rate, fading range and fade- 
duration distribution. 

(6) Radio receiver noise level, combined with the receiving- 
antenna system noise power, the radio transmitter -output 
noise and the intermodulation noise, all referred to the radio 
receiver input terminals. 

(7) Diversity gain, both theoretical and measured. 

(8) Message-channel signal-to-noise power ratio, S /N , at 

, ,. . oc oc 

the radio receiver output. 

(9) The relationship between S /N at the radio receiver out- 

oc oc 
put and the message error rate of the received information. 

From a study of the above definable and measurable parameters 

of a radio frequency communication system it is possible to determine 

the performance of various types of systems. Direct comparisons can 

then be made between the various systems at common points within 

the systems, in order to obtain relative figures of merit. 

2. 2 Design and Testing of Equipment and Systems 

The principal requirement, which may be used as a guide when 
designing and testing radio communication systems, is that the aver- 
age (received) message error rate should not exceed a specified value 
for specified message-load conditions on the system. This allowable 
average message error rate determines a required time-average 



-8- 

signal-to-noise ratio at the message-signal decoder-unit input. This 
required time-average signal-to-noise ratio at the message-signal 
decoder-unit input may be used as a starting point in the design of the 
radio frequency section of a communication system. It is understood 
that the signal level, at the input to the message -signal decoding unit, 
is required to be above a particular minimum level for proper oper - 
ation of the decoding unit; proper amplification will maintain this 
signal level. However, the signal-to-noise ratio cannot be enhanced 
by amplification. 

The design of each type of radio communication system can 
be carried out for various message-error rates and message loading; 
the different types of communication systems can then be compared 
for similar grades of service in terms of factors such as radio fre- 
quency power required at the transmitter output, required radio fre- 
quency spectrum, total cost of system, operational complexity, etc. 

The performance and the cost of a radio communication system are 
governed largely by the radio-wave transmission loss, fading range, 
fading rate and the amount of frequency selective fading of the trans- 
mitted radio frequency signal. However, it is convenient to design 
the system for a non-fading or steady radio-frequency signal and then 
estimate the system performance with a fading type of radio frequency 
carrier signal; this is the procedure to be followed in this work. This 
procedure is feasible under conditions where the fade rate of the re- 
ceived carrier signal is low compared with either the duration of the 
binary pulse or the lowest frequency in the message signal. Methods 
of estimating the performance of radio communication systems, based 
on the above principles, are outlined in Sections 6 and 7 and Appendi- 
ces B. C, and D. 



-9- 
3. MODULATOR-TRANSMITTER PERFORMANCE 

The transmitter -output RF-signal characteristics, such as re- 
quired spectrum bandwidth, B , average power level, P , and time- 

ri t 

amplitude distribution of the envelope, depend upon the type of modu- 
lation and the type of modulating signal. In order to obtain useful 
results it is necessary to derive the relationships between the average 
power level of the modulating signal, P , and the characteristics of 
the RF signal. For convenience, a sinusoidal modulating signal, with 

a power level P at the Modulator input, is used as a "standard" of 
ms 

comparison. Proper or equivalent modulating- signal power levels, 

P , for noise -type modulating signals are then obtainable relative to 

P by means of the measurable or the specified characteristics of 
ms 

the Modulator. For amplitude modulation, the Modulator character- 
istics are in terms of P versus the amplitude -modulation index, 

ms 

m ; for frequency modulation, P versus the carrier-frequency devi- 

a n 3 ms n ' 

ation, AF, would be required. The amplitude -modulation index, m , 
may vary from to 100% without serious intermodulation problems; 
modulating- signal power levels greatly in excess of those required for 
100% amplitude modulation will yield excessive intermodulation noise, 
and should not be used. For frequency modulation, the modulating - 
signal power level, P , determines the RF-signal spectrum band- 
width, B r , and hence the chosen level for P depends upon the choice 
rf m 

of B and the FM Modulator characteristics, 
rf 

In this section quantitative relationships are derived between: (a) 

the modulating -signal average power level, P , (b) the RF signal 

bandwidth, B r , and (c) the distribution of the time -varying envelope of 
rf 

the transmitter -output RF signal. The results apply to both a sinus- 
oidal modulating signal and a white -noise type of modulating signal. 
SSBSC-AM and FM systems are considered in this analysis. 



-10- 
3. 1 Required Radio Frequency Signal Spectrum 
Bandwidth for Radio Communication Systems. 
The required radio-frequency signal spectrum bandwidth depends 
upon the bandwidth of the modulating baseband signal and the type of 
modulation employed. In turn, the baseband- signal bandwidth depends 
upon the number of information or message channels, the bandwidth of 
each channel, and the type of multiplexing used to form the baseband- 
signal spectrum. The problem can be simplified somewhat by con- 
sidering the radio-frequency signal bandwidth required in terms of a 
given baseband- signal bandwidth. This procedure makes possible a 
direct comparison of the radio-frequency signal bandwidth require- 
ments for various types of radio communication systems, with the 
same baseband signal, or message load. The separate problem of 
minimizing the baseband signal bandwidth required, for a given 
number of information channels or message load, involves a consider- 
ation of the principles of encoding [ Nyquist, 1924, 1928; Hartley, R. 
V. L. , 1928] and multiplexing techniques [ Landon, 1948] and is not 
considered in this paper. 

3. 1. 1 Radio Frequency Bandwidth Required for SSBSC-AM Systems 
In an SSBSC-AM (single sideband suppressed carrier, amplitude 
modulated) system the required radio frequency signal bandwidth is 
approximately equal to the bandwidth of the composite baseband or 
modulating signal. 

3.1.2 Radio Frequency Bandwidth Required for FM Systems. 

In an FM system the required radio frequency bandwidth depends 
upon the baseband or modulating- signal bandwidth, the modulation 
index, and also the amplitude distribution of the modulating signal, as 
will be shown later. The required radio-frequency signal bandwidth 



-11- 

for FM systems is usually greater than the modulating- signal band- 
width, by a factor ranging from 2. 5 to 25, or more. 

3.1.2. 1 Required FM Radio Frequency Signal Bandwidth 
for a Single Sinusoidal Modulating Signal. 

When a single sinusoidal modulating signal, of frequency f , is 
employed to frequency-modulate a (radio-frequency) carrier signal, 
the resultant modulated-carrier power spectrum (theoretically) con- 
sists of an infinite number of components [ Carson, 1922, 1929; Van 
der Pol, 1930; Hund, 1942] . For our purpose we restrict the 
required or transmitted radio-frequency signal spectrum bandwidth so 
as to include only the "significant" spectral components; that is, we 
arbitrarily include only those components having amplitudes equal to 
or greater than one percent of the amplitude of the unmodulated car- 
rier-signal amplitude. Under these conditions the "required" radio- 
frequency signal bandwidth B is given by 

rf 

B . = f x<j> (AF/f ) (3. 1) 

rf c c 

where, f = Frequency of the sinusoidal modulating signal, 

in c/s. 

AF = Peak deviation of the radio-frequency carrier 

signal from its un-modulated frequency, in c/s. 
This is the frequency deviation above and below 
the carrier-signal frequency. 

cj> (AF/f ) = Function of AF/f , which gives the number of 
significant sidebands in the radio-frequency 
spectrum. 

For a given value of AF, corresponding to a particular value of 

modulating- signal average -power level, P , the radio-frequency 

bandwidth increases with increasing values of f . Hence, the maximum 

° c 

value of B P is determined for the case where the frequency 
rf 



-12- 



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-13- 
of the sinusoidal modulating signal is equal to f , the highest fre- 
quency in the modulating baseband signal. Therefore, throughout this 
report the required (maximum) radio-frequency signal spectrum band- 
width, B ., is derived from, 
rf 



B , = f x <j> (AF/f ) (3.2) 

rf m m 



where, f = Highest frequency in the modulating baseband 

signal, in c/s. 

The function (j> (AF/f ) is shown plotted in figure 3-1 for both a 
sinusoidal modulating signal and for a "noise-type" modulating signal; 
the latter case is discussed in Subsection 3.2.3. Note that a smoothed 
curve is drawn for the case of a sinusoidal modulating signal. 

The decision to include only the RF spectral components having 
amplitudes greater than one percent of the unmodulated radio-fre- 
quency carrier signal, is based on measured results such as shown 
in figure 3-2. The curves in figure 3-2 indicate the effects of band- 
width clipping of the radio-frequency spectrum. These curves also 
show the performance of an FM receiver, in terms of the receiver- 
output signal-to-noise ratio, S /N , for a received radio-frequency 

oc oc 

signal at and above the "threshold level" of the receiver. At thres- 
hold (or below), the receiver IF bandwidth, B , should be somewhat 
greater than the radio-frequency spectrum bandwidth, B .; as given 
by (3.2), or the curve in figure 3-1. Above threshold, the receiver 
IF bandwidth may be somewhat less than B , without seriously de- 
grading the receiver output signal-to-noise ratio. Hence the use of 
(3.2) and figure 3-1 results in system performance estimates which 
are conservative above the receiver threshold level and are slightly 
optimistic below the receiver threshold level. 



14- 



RECEIVER OUTPUT S oc /N c RATIO VERSUS B r f/Bjf FOR 
VARIOUS RECEIVER- INPUT TOTAL RF SIGNAL POWER LEVELS 
STEADY RECEIVER -INPUT SIGNAL 
SINUSODIAL MODULATING SIGNAL 



-5 



-10 



.a 



o 
o 



-15 



o 



-20 



-25 



-30 



-40 



1^-2l 




RECEIVER INPUT- SIGNAL POWER 
RELATIVE TO THRESHOLD LEVEL, db 



0.5 



1.0 



1.5 



2.0 



25 



10 



B r f _ RF SPECTRUM BANDWIDTH 
B if BANDWIDTH OF RECEIVER IF 

Figure 3-2 






-15- 
3.1.2.2 Required FM Radio Frequency Signal Bandwidth 

for Various Types of Modulating Signals. 

In the preceding analysis the maximum value of the required 

radio-frequency signal bandwidth, B , was derived on the assumption 

that the modulating signal was sinusoidal, with a frequency f , equal 

m 

to the top frequency in the baseband, and that all of the modulating- 
signal power was concentrated in this one signal. Under these con- 
ditions the radio-frequency signal spectrum and the required band- 
width, B f , will be a maximum for a given set of system parameters 
and for a single sinusoidal modulating signal of frequency f 

We wish to determine the relationship between the required radio- 
frequency signal bandwidth, B , for a single sinusoidal modulating 

n s 

signal at the top baseband frequency, f , and the required radio- 
frequency signal bandwidth, B . , for a composite modulating signal. 

rf n 

The spectral components of the composite modulating signal are 

assumed to be distributed uniformly throughout the baseband modulat- 

ing-signal spectrum and the cumulative amplitude distribution of this 

signal may be of various types. Radio-frequency signal bandwidth 

comparisons are made on the assumption that f is the highest f re- 
in 

quency in the composite modulating signal. 

The radio-frequency carrier- signal frequency deviation, AF, 
varies directly with the amplitude of the (voltage) envelope of the com- 
posite modulating signal, and hence the cumulative distribution of the 
envelope of the modulating signal is used to determine A F on a sta- 
tistical basis. That is, the "short-term" values of A F can be deter- 
mined for various percentages of time, including the maximum or 
peak value of A F which occurs during voltage or power peaks of the 
modulating signal. It should be noted that the short-term period is 
assumed to be short compared with the time variations of the envelope 
of the modulating signal. 



-16- 
The short-term value of the required radio-frequency signal 

spectrum bandwidth B , is determined by both the deviation ratio 

rf 

A F/f and f ; where f is the short-term frequency of the modulat- 
c c c 

ing signal. The time included in the short-term period is assumed to 

be large compared with one cycle of the modulating frequency; during 

this period, AF and f are assumed to be constant. However, the 

c 

values of A F and f are assumed to vary independently, with time. 

The radio-frequency signal bandwidth, B „ , required for a com- 

rfn 

posite-type of modulating signal, is defined here as the radio-fre- 
quency signal spectrum bandwidth containing 99. 99 percent of the 
spectral energy of the modulated RF carrier signal. This definition 

of B r is equivalent to the above definition of B - , for the case 
rfn ris 

where the modulating signal is a single sinusoid. However, this 
bandwidth is exceeded for a small percentage of the time, during 
which the (power) peaks in the composite modulating signal exceed 
the level required to generate a radio-frequency signal spectrum 
greater than B . . Furthermore, these modulating -signal power- 
level peaks, which are required to generate radio-frequency signal 

spectrum bandwidths exceeding B r , will also depend upon the simul- 

rfn 

taneous short-term frequency of the modulating signal. 

From the above considerations it is evident that both AF and 

s 

AF are the peak values of the short-term AF; while AF is seen to 
n 

be proportional to the (voltage) envelope of the modulating signal. 



3. 1.2.3 Required FM Radio Frequency Signal Bandwidth for a 

White -Noise Modulating Signal. 

Reliable information is scarce on the relationship between the 
required radio-frequency signal spectrum bandwidth, B f , and 
A F/f , for modulating signals composed of groups of sinusoidal 



-17- 

signals arranged in various frequency combinations in the modulating - 

signal baseband spectrum. However, the relationship between B 

rfn 

and A F /f has been calculated and measured bv Medhurst [ 19561 
n m 7 l / j 

for the case where 2tt (f -f ) equal- amplitude, randomly-phased sinus- 

m o 

oidal signals were assumed to simulate white-noise across the base- 
band; this artificial white-noise signal was used in his calculations 
as the composite modulating signal, where f was the lowest frequency 

and f was the highest frequency in the modulating signal. The results 
m 

of his work are shown plotted in figure 3-1, see "noise modulation" 

curve; see also [ Middleton, 1951; Stewart, 1954] . 

In figure 3-1, AF is defined as the peak value of the frequency 

deviation of the carrier signal. When a sinusoidal modulating signal 

is used, AF is constant, and is proportional to the peak amplitude 
s 

of the modulating- signal voltage. For noise-signal modulation, AF 

is proportional to the "peak amplitudes" of the voltage envelope of 

the noise -modulating signal; these peak amplitudes are exceeded for 

only . 04 percent of the time, and are estimated to be 11 db above the 

average power value of the noise modulating signal. [Gladwin, 1947; 

Alversheim and Schafter, 1 952; Medhurst, 1956]. AF should be 

n 

considered as a peak deviation which is 11 db greater than the "rms 

AF " frequency deviation. The rms AF frequency deviation deter- 
n n 

mines or generates the RF signal spectrum having a bandwidth, B . 

Both AF and rms AF are quantitatively defined (and measured) so 
n n 

as to make proper allowances, for the fact that the short-term devi- 
ation, AF, and the short-term modulating- signal frequency, f , vary 
with time for noise-signal frequency modulation. 

The following derivations are for the purpose of obtaining the 
relationships between the frequency deviations, AF and AF , and 
the modulating -signal average-power levels, P and P ; for the 

6 6 e r ms mn 

conditions: 



-18- 



B , = B , = B (3.3) 

rf rfs rfn 



and, 



f (for sinusoidal-signal mod. ) = f (for noise-signal mod. ) 

(3.4) 

The above factors are defined as follows: 

AF = Peak deviation of the RF carrier signal for a sinusoidal 
modulating signal, in c/s. 

AF = Peak deviation of the RF carrier signal exceeded for 
n 

only . 04 percent of the time; for a white-noise modu- 
lating signal, in c/s. 

P = Average long-term power level of the sinusoidal modu- 

lating signal; measured at the Modulator -input terminals, 
with a true rms reading meter. 

P = Average long-term power level of the noise modulating 

mn . , 5 5 , *\ • . , . . 6 

signal; measured at the Modulator-input terminals, 

with a true rms reading meter. 
The advantage of referring our results to sinusoidal- signal modu- 
lation conditions is due to the fact that B . , A F , and P may 

rfs s ms 

either be calculated or may be measured conveniently with standard 

types of test equipment; these measured factors may then be used as 

reference levels. It is inconvenient to calculate or to measure B - , 

rfn 

AF , and P , because AF can only be measured indirectly in 
n mn n 

terms of B , . 
rfn 

Figure 3-3 shows the "relative levels", in db units, associated 

with B ., AF, and P , for sinusoidal-signal and for white-noise- 
rf m ° 

signal frequency modulation. From figure 3-3, 

10 Log (AF /AF ) = 10 Log. _ (P /p ) + 8 (3.5) 

10 n s to l mn ms v 

Values of the factor 10 Log, . (iiF /A F ), for particular values of 

10 n s 

B Ji , were obtained from figure 3-1 and substituted in (3. 5) to obtain 
rf m ° v ' 

corresponding values for 10 Log n .(P /P ); the results of .these 

10 ms mn 

calculations are shown in figure 3-4. 



19' 



RELATIVE LEVELS OF AF AND P m , SINUSOIDAL AND NOISE 
MODULATING SIGNALS AND FOR EQUAL RF SPECTRUM BANDWIDTHS 



AF n , db t 



NOTE , B r f s = B r f n = B rf 



f = {(rmsAF n ) + ll}-{(rms 
r „-AF t ),db^ j x _, , 

ROM Fig. 3-D I. s \ p mn + "/ \ p ms + 3 J » 



(AF, 

(FROM Fi 



AF S ) +3} , 
db 



db 



3 db 



AF S , db 



rms AF S , db 
OR 



rms AF n , db 

OR 
p mn. dbm 



II db 



SINUSOIDAL SIGNAL 
MODULATION 



NOISE SIGNAL 
MODULATION 



Figure 3-3 






20- 



30 



RF SPECTRUM BANDWIDTH, B r f, 

VERSUS RELATIVE MODULATING-SIGNAL POWER LEVELS, 

FOR FREQUENCY MODULATION 

i — I 1 — i — i 1 — i — i r~ 



25 



< 

CD 



3 

cr 
h- 
o 

UJ 

o. 

CO 

u_ 
cc 

Q 
UJ 
CC 

3 

o 

UJ 

cc 



P ms = AVERAGE POWER LEVEL OF THE TOTAL 
SINUSOIDAL MODULATING SIGNAL. 



WHITE-NOISE MODULATING SIGNAL 



NOTE , B rfs = B rfn = B r f 



CO 
C3 



20 



3 
Q 
O 



2 15 

>- 
o 

z 

UJ 

3 

o 

ui 
cc 
u. 



tj E 
cor- 




4 



10 LOG 



ms 



io\P, 



mn> 
Figure 3-4 



db 



-21- 

It should be noted that the RF spectrum bandwidth, B ., is either 

rf 

given or it may be chosen by the system designer. Also the highest 

modulating frequency in the modulating signal, f depends upon the 

m 

number and the type of message channels and also the type of multi- 
plexing equipment being used. Hence the value of B Ji is available 

rf m 

for use in figures 3- 1 and 3-4. 

Figure 3-1 may be used to determine values for A F /f and 

s m 

AF /f ; for the conditions given by (3.3) and (3.4). The AF vs 
n m ° ' * s 

n/P characteristic of the (frequency) modulator should be linear 

ms 

for values of A F equal to A F , for sinusoidal- signal modulation; or 

equal to AF , for a white -noise modulating signal. If the modulator 

characteristic is non-linear for values of A F , obtained from figure 

3-1, intermodulation "noise" will be added to the system noise, and 

the signal-to-noise ratio at the receiver output will be degraded. 

Figure 3-4 may be used to obtain correct values for P /P 

ms mn 

which correspond to particular values of B /f ; assuming the con- 
ditions given by (3.3) and (3.4). The sinusoidal-signal modulating- 

power level, P , which is required to produce a frequency deviation 
ms 

of the RF carrier signal, AF , and an RF spectrum bandwidth, B, , 

° s frs 

may be obtained from the measured AF vs n/P characteristic 

s ms 

of the frequency modulator. The required noise -signal modulating - 

power level, P , may then be obtained from figure 3-4. If pre- 
mn 

emphasis is used, the pre-emphasis circuit must be inserted at a 

point in the system which follows the modulator-input terminals; 

note that P and P are measured at the modulator -input terminals 
ms mn 



It is common practice to estimate B . from, 

r rf 



B = f x z( ^ + 1 
rf m V f 

m 



-22- 

The required bandwidth obtained from (3.6) will include only the 

RF spectral components having amplitudes which are approximately 

10 percent of the unmodulated RF carrier- signal amplitude, or greater. 

More precisely, 99 percent of the RF- spectrum energy will be within 

the bandwidth determined from (3.6). Hence the use of (3.2) and 

figures 3-1 and 3-4 yield required-bandwidth values, B , (99.99%) 

rf 

which are somewhat greater that the B A 99 %) bandwidths, which would 

rt 

be obtained from (3.6). The following relationship between B f (99. 99% 

bandwidth) and B A 99 % bandwidth) has been determined, using informa- 
rf 

tion similar to that used to obtain the curves in figure 3-1 (see Medhurst, 
1956): 



<t> 



AF 



m 



99. 99% bandwidth J « 1. 4 



m ' 



99 % bandwidth 



or, 



B ( 99. 99% bandwidth J » 1. 4 B J 99% bandwidth J 



(3.7) 



The use of the smaller RF spectrum bandwidth, obtained from (3.6), 
will result in a degradation of the receiver-output signal-to-noise ratio. 
The amount of this degradation is shown in figure 3-2. 






-23- 
3.2 Radio-Frequency Signal Time -Amplitude Characteristics 

for Radio Communication Systems 

The time -amplitude characteristics, or the distribution of the 
envelope of the transmitter-output RF signal, depend upon the statis- 
tical characteristics of the modulating baseband signal and the type of 
modulation. Other considerations relating to the baseband signal are 
the same as discussed in section 3.1. 

3.2. 1 Radio-Frequency Signal Time -Amplitude Characteristics 

for SSBSC-AM Systems 

An SSBSC-AM Modulator translates the baseband- signal spectrum 
to an RF- signal spectrum. In an ideal SSBSC-AM system, wherein 
there is no modulator distortion, the distribution of the RF signal is 
identical to the distribution of the baseband signal. Furthermore, the 
average transmitter -output power level, P , is related to the average 
power level of the modulating signal, P , in terms of the combined 
characteristics of the modulator and the transmitter. For maximum 
modulation efficiency and to limit over -modulation effects, the aver- 
age power level of the modulating signal, P , should be adjusted so 
as not to exceed a short-term amplitude -modulation index, m , of 
100% for more than a given percentage of the time. 

3.2.1.1 Radio-Frequency Signal Time-Amplitude Characteristics 
of SSBSC-AM Systems for a Single Sinusoidal Modulating Signal 

When a single sinusoidal modulating signal of frequency, f , and 

power level, P , is employed in an SSBSC-AM system to amplitude - 
ms 

modulate a (radio-frequency) carrier signal, the resultant transmitter- 
output RF signal consists of a single (sinusoidal) RF signal. 



-24- 

The average power level of the sinusoidal modulating signal re- 
quired for 100 percent amplitude modulation, P (100%), maybe 

obtained from the modulator characteristics, in terms of P versus 

ms 

percent amplitude -modulation, m ; the modulator characteristics may 

either be specified or measured. The above value of P (100%), for 

ms 

100 percent amplitude modulation, may be used as a "limiting reference 
level" for the average power of the modulating signal- -for various types 
of modulating signals. From a practical viewpoint, it is convenient 
to estimate or to measure the performance of an amplitude modulator 
using a sinusoidal modulating signal. 

3.2. 1.2 Radio-Frequency Signal Time-Amplitude Characteristics 
of SSBSC-AM Systems for a White-Noise Type of Modulating Signal 

In Section 3.2.1.1 it was noted that the maximum value of (sinus- 
oidal) modulating- signal average power, P , (100%), was obtainable 

ms 

from the SSBSC-AM modulator characteristics, on the assumption 
that the modulating signal was sinusoidal and that all of the modulating- 
signal power, P , was concentrated in this one signal, 
ms ° 

We wish to determine the average power level, P , of a com- 

mn 

posite (noise-type) modulating signal which will insure that 100 percent 

amplitude modulation of the RF signal is exceeded for no more than a 

specified percentage of the time. The noise-type modulating-signal 

average power level, P , is to be referred to the previously-deter- 

mn 

mined sinusoidal modulating-signal power level, P (100%). 

mn 



-25- 



The amplitude of the radio-frequency signal varies directly with 

the amplitude of the (voltage) envelope of the composite modulating 

noise-type signal, for values of modulating- signal power less than the 

"100 percent modulation" level, P (100%). Hence, for noise-signal 

modulation the " 1 00 percent modulation peaks", P , correspond to 

mnp 

particular "peak amplitudes" of the voltage envelope of the noise-modu- 
lating signal; these peak amplitude values, P , may be defined in 

mnp 

terms of P (100%) and also in terms of the average power level of 
mn 

the noise -type modulating signal, P 

For a white-noise signal, the envelope of the signal will be 11 db 
or more above the average power level, P , for . 04 percent of the 
time [Gladwin, 1947; Alversheim and Schafter, 1952; Medhurst, 1956] . 

Since the peak value of the sinusoidal modulating signal for 100 

percent amplitude modulation, P (100%) is 3 db above its average 

msp 

value, P (100%), and P (100%) has the same level as P (100%), 

ms msp mnp 

it follows that the average sinusoidal modulating- signal power level, 
P (100%), is 8 db higher than the average power level of the noise- 
modulating signal, P --for the condition that the noise -modulating 

mn 

signal will exceed 100 percent amplitude for . 04 percent of the time. 

Following the above line of reasoning, combined with the distribu- 
tion of a noise-type signal [ Plush, R. W. , et al, 19&0] , the ratio, 

P (100%)/P , was calculated for a range of percentage-of-time 
ms mn 

during which amplitude modulation exceeded 100 percent. The results of 
these calculations are shown in figure 3-5. 



-26- 



AMPLITUDE MODULATION EXCEEDING 100 PERCENT VERSUS 
RELATIVE MODULATING -SIGNAL AVERAGE POWER LEVELS, 

FOR SSBSC-AM 





99 




98 




95 




90 


Q 




3 




j z 


80 


:* u 




< oc 


/0 


UJ 




•- O. 




z 


60 


So 




a: UJ 


50 


uj R 




0. UJ 




UJ 


40 


o y 




o x 




— UJ 


30 


o — 




i z 


?0 


$ o 




15 




tr 3 


10 


3 a 




Q o 




uj 2 


5 



0.5 

0.2 

0.1 

005 

0.01 













































P ms ( 100%) = AVERAGE POWER LEVEL OF TH 
SINUSOIDAL MODULATING SIGNAL F 


E TOTAL 

EQUIRED_ 

N 










FOR 100% AMPLITUDE MODULATIO 








P mn = AVERAGE POWER LEVEL OF THE 

TOTAL WHITE-NOISE MODULATING SIGNAL 




























































































































































































































































6 -< 


-\ 


! ( 


) i 


i 


i 


; t 


i 


3 12 



iolog io d ss =mmr, l0 [ p ms( |00%) ] t db 



' mn 
Figure 3-5 



-27- 
Figure 3-5 may be used as a guide in determining the (approximate) 

proper average power level, P , for the noise-type modulating signal, 

in terms of the measurable average power level, P , of a sinusoidal 

ms 

modulating signal, and the percentage of time that 100% amplitude - 

modulation conditions are exceeded. However, figure 3-5 does not 

yield accurate estimates of the intermodulation power generated within 

the Amplitude -Modulator, due to over-modulation; this work should be 

extended. 

The factor, D = P (100%)/P , in the abscissa scale of 
ss ms mn 

figure 3-5, is required in the design or the performance-testing of 
SSBSC-AM systems, for proper adjustment of the modulating- signal 
power level, at the modulator input terminals. 

3.2.2 Radio-Frequency Signal Time -Amplitude Characteristics 

for FM Systems. 

In an FM system the amplitude of the transmitter -output RF 
signal is independent of the characteristics of the baseband modulating 
signal. 

3. 3 Peak Power Requirements for SSBSC-AM Systems 

In AM (amplitude -modulated) systems, the transmitter-output 

short-term peak power levels, P , should be related to the transmit - 

— — tp 

ter-output average power level, P , in precisely the same manner 

that the short-term peak power levels, P , of the composite modu- 

— — mnp 

lating signal, are related to the average total power level of the modu- 
lating signal, P . These relationships are required in the transmit - 
mn 

ter output in order to avoid excessive intermodulation distortion in the 
transmitter -output RF signal during periods of peak modulating power. 



-28- 
The above peak-power capability requirements for the transmitter 

depend directly upon the amplitude -time distribution of the envelope of 
the composite baseband modulating signal. A composite modulating 
signal which is composed of a number of independent voice-message 
signals combined with teletype -message and data-phone message 
signals, may be assumed to have a distribution which approximates 
the distribution of a white-noise signal. For a white-noise signal 
having a normal distribution, the peak power exceeded for one percent 
of the time is 7.3 db above the average power level of this signal. 
Other combinations of modulating signals, such as 8 or more equal- 
amplitude randomly-phased sinusoids, have distributions [Slack, 1946] , 
[ Plush, R. W. , et al, I960] in which the composite-signal peak- 
power-level amplitude, P , exceeds the average power level, P , 

mnp m 

by only 8 db for one percent of the time. Estimates of the percentage 

of time during which the transmitter -output short-term power exceeds 

its average power level, P , by 1 1 db, may be obtained from figure 3-5. 

One method of qualifying the transmitter characteristics is to 

specify that the transmitter (power amplifier) must be linear to within 

a particular degree, usually one percent, for transmitter-output-power 

peaks, P , approximately 1 db above the average power -output level, 
tp 

P , of the transmitter. 



-29- 

4. RADIO RECEIVER- COMBINER PERFORMANCE 

This section deals with the performance of the radio receiver con- 
sidered as a unit in the radio communication system. The method of 
analysis is general and may be applied to any type of receiver, pro- 
vided that the steady received- signal performance characteristics of 
the receiver are available, either measured or calculated. 

In the following work, the receiver performance characteristics 
are first determined for a steady received signal. The receiver per- 
formance, that is, the receiver output signal-to-noise ratio, is then 
calculated for a time -varying received signal by combining the steady- 
signal receiver characteristics with the cumulative distribution of the 
received radio -frequency signal. The effects of various orders of 
diversity on the resultant signal-to-noise ratio in the combined base- 
band signal are included in the analysis by combining the steady- signal 
characteristics of the radio receiver with the performance character- 
istics of the combiner; at this point, there may be a choice between 
predetection and post detection combining. 

Radio receiver performance is defined here in terms of the re- 
lationship between the receiver-output signal-to-noise ratio, S /N , 
and the following factors: 

(1) receiver-input radio-frequency signal total available power, 

P ; see Appendix A for definition of available power, 
r 

(2) available receiver-input noise power, N ; which is composed 
of the noise power received by the antenna plus the noise 
power which originates in the receiving-antenna system, 

(3) noise power contributed by the radio receiver, (F - l)kT B 

= kT B. r and referred to the receiver input, 
er if 

(4) the power level of the modulating signal, P , 

(5) the signal-power loading ratio, P /p ., in the modulating 
baseband signal message channel. 



-30- 

(6) message -signal bandwidth, B , and 

(7) the highest frequency in the modulating baseband signal, f 
The performance of a radio receiver, as defined above, depends 

upon the type of modulation being used and also depends upon the par- 
ticular individual characteristics of the receiver. The required radio- 
frequency total signal power, P , at the receiver input, is directly 
related to the total noise power at the same point, (see Appendix A). 

The nature of the radio receiver output signal-to-noise ratio is 
dependent upon the statistical characteristics and the median power 
level of the radio-frequency signal at the receiver input; for example, 
the characteristics of the output noise of a FM receiver, when operat- 
ing in its threshold region, differ considerably from the output noise- 
power characteristics above the threshold region. The message error 
rate, for a given receiver output signal-to-noise ratio, will depend 
upon the characteristics of the noise at the receiver output, hence this 
factor is important when considering the overall performance of a 
radio communication system. 

The above-listed factors (1) to (7) inclusive, are treated in Section 
4. Detailed methods of applying the above analysis, to determine the 
radio receiver output signal-to-noise ratios, are given in Subsections 
4.5 and 4.6; in which are developed quantitative performance character- 
istics in graphical form, for SSBSC-AM and FM receivers, respectively. 
This work includes steady received-signal and Rayleigh-fading re- 
ceived-signal performance curves for non-diversity, dual diversity, 
and quadruple diversity. Maximal- ratio type of combining is assumed. 

4. 1 Radio-Frequency Signal Characteristics and Power Levels in the 

Receiving System. 

The radio frequency signal power level at the receiver input, P , 
is a key factor in the performance of a radio communication system; 
this condition applies for either a steady or a time-varying received 



-31- 
signal. The power which the radio transmitter must provide is directly- 
related to the power P required at the radio receiver input through 
the total transmission path loss. Furthermore, the performances of 
various types of radio receivers are best compared in terms of the 
receiver-input RF signal power levels required for the same value of 
receiver -output signal-to-noise ratio; this method of comparing 
receivers determines the receiver "sensitivity" and makes the proper 
allowance for the receiver noise figure and the receiver IF noise- 
power bandwidth. In practice, it is also found to be more convenient 
to measure the receiver-input RF signal power than to measure the 
pre-detection signal -to -noise ratio. This situation is particularly 
apparent to the system-testing engineer when measuring a time-vary- 
ing received signal, and for cases where automatic gain control (age) 
and/or limiting is used in the receiver. 

The following characteristics of a time-varying received radio- 
frequency signal are of importance, insofar as they affect the receiver 
output signal-to-noise ratio: 

(a) Median power level, for the sampling-time or sampling-period 

(b) Cumulative distribution of the envelope of the received radio- 
frequency signal 

(c) Fade rate at the median power level 

(d) Time variability of the amplitude and phase cross-correlation, 
between the spectral components of the received radio fre- 
quency signal- -due to "selective fading" effects and multi- 
path radio-wave propagation conditions. 

The effects of the above listed received- signal characteristics (a), 
(b), and (c) are considered later in this paper, to estimate radio re- 
ceiver performance for a time-varying radio-frequency signal. For 
tropospheric-propagated radio-frequency signals, (a) may be calcu- 
lated [Rice, Longley and Norton, 1959] , (b) has been extensively 
measured and has been found to closely approximate a Rayleigh distri- 
bution, and (c) has been found to vary between approximately 0. 1 cps 
to 1 cps. 



-32- 

Additional experimental work is required to determine an 

accurate measure of the relationship between characteristic (d ) and 
the radio receiver output signal-to-noise ratio; at present, experi- 
mental measurements of this type are very meager. In cases where 
relatively wide radio -frequency spectra are used, it is necessary to 
consider the non-correlated amplitude and phase variations between 
the signal components across the radio frequency and the baseband 
spectrums. These factors influence the receiver output signal-to- 
noise ratio and the message error rate and are usually involved in the 
transmission characteristics of the radio-wave path. The path-band- 
width cross -correlation or covariance factor is probably a function of 
the beam width of the antenna -pattern, length of path,' and carrier fre- 
quency. Measurements of the tropospheric radio-path bandwidth capa- 
bility have been made by Clutts, Kennedy and Trecker [ 19&0] on a 
185 mile path between Florida and Cuba. 

From theoretical work by Staras [1955] we have the following 
relationship: 

P = ^3— Mc/s (4.1) 

where d = radio wave transmission path length in hundreds 

of miles 

(3 = correlation (radio frequency) bandwidth in Mc. 
By definition, the cross -correlation coefficient 
of the radio-frequency spectral components 
spaced (3 Mc/s is 0.5. 

Equation 4. 1 is probably conservative and hence can be used to 
obtain an estimate of the usable radio frequency bandwidth in cases 
where antenna beam widths are equal to or greater than the angle sub- 
tended by the effective "scatter volume". The usable radio frequency 
bandwidth is much greater for cases where the antenna beam width is 



-33- 
considerably smaller than the angle subtended by the scatter volume. 

Where the antenna beams are of the order of 1 degree or less a some- 
what larger value of usable radio frequency bandwidth could be assumed. 
Narrow-beam antennas reduce the multi-path effects and thereby result 
in an increase of usable radio frequency bandwidth; see section 4. 3. 

4. 2 Pre-Detection Noise-Power Levels in the Receiving System 

The noise performance of the receiving system depends upon the 
amount of noise power which is contributed by the various units in the 
receiving system, and which subsequently interferes with the pre-de- 
tection received signal, P. f . In this analysis an estimate is made of 
the total noise power contributed by the entire receiving system, from 
and including the receiving antenna to the pre -detection point in the 
radio receiver. 

In order to make the results of this analysis convenient for use in 
system design work and performance tests, the total of the noise power 
contributed by the receiving system is. referred to the radio receiver 
input, in terms of the receiving-system parameters. Using this pro- 
cedure, the system designer can accurately estimate the level of the 
noise power at the receiver input; this same noise-power level can be 
conveniently measured, when performance tests are made, by using a 
calibrated (linear) receiver—where the receiver is calibrated in terms 
of receiver -input RF signal power versus receiver IF-output signal 
power or, the receiver age voltage. Only amplitude -type of noise 
power is considered, as distinguished from phase or frequency-modu- 
lation noise; the latter is measurable only in the output of an FM 
receiver, and does not appear as amplitude noise in the predetection 
circuits of the receiver. Post-detection noise and its effects on the 
receiver-output signal-to-noise ratio is considered in section 4. 3. 



-34- 

In this work all noise powers in the receiving system are added 
algebraically on an average power-level basis, as measured by a true 
rms-reading meter with an appropriate time constant. The noise 
power developed within the receiving-system units is assumed to be 
"thermal" type noise and to have a flat power spectrum over the band- 
width accepted by the receiver. Interference from other radio trans- 
missions, which will introduce an effective noise component designated 
as N .in Appendix A, are excluded from the present analysis, since 
their inclusion would require a consideration of the statistical charac- 
teristics of the interfering signal. This task is beyond the scope of 
this report. 

The transmitter -output "noise" power N , associated with the 
received RF signal, is composed of "transmitter amplitude noise" and 
"transmitter intermodulation noise. " 

Transmitter amplitude noise usually originates in the transmitter 
output stages; this noise is present even though the transmitter is not 
being modulated, and may be measured, relative to the un-modulated 
RF carrier signal, at the transmitter output. It may be measured by 
connecting a transmitter and a receiver back-to-back through an at- 
tenuator, and measuring the receiver baseband -output noise power, 
under conditions of no modulation on the transmitter carrier signal. 
The receiver-output total noise, N , (see figure 4-A) is measured 
for a range of receiver-input carrier-signal powers and the results 
plotted. The relative value of transmitter noise is indicated at that 
portion of the curve where the receiver-output noise does not decrease 
with further increase in the received carrier-signal power. The trans- 
mitter-output signal-to-noise ratio should be high enough to insure that 
the "leveling-off" portion of the curve will be above the operating or 
required RF signal-power levels of the receiver performance charac- 
teristic. 



-35- 

The net effect of transmitter -output noise is to place an upper 
limit on the receiver -output signal-to-noise ratio, and is similar to 
the effects of (a) the signal-to-noise ratio at the modulator input 
terminals, (b) equipment intermodulation noise, and (c) path-modulation 
effects. The latter subjects are discussed in section 4.3. 

The conclusions to be drawn from the above discussion of the 

transmitter-output amplitude noise, N , are that proper modifications 

at 

for the effects of this noise power can be made on the system design 

curves and also on system-performance test results, and that N 

at 

need not be considered when calculating the noise levels or the per- 
formance of the receiving -antenna system. Consequently, N does 
not appear in the expression for the receiving-system available noise- 
power level, N , or effective noise temperature, T , at the receiver- 

J- J\. 

input terminals; see figure 4-1 and Appendix A. 

Following is an outline of the method used to estimate the total 
(average) available noise-power level, N , at the (radio frequency) 
receiver input. Referring to figure 4-1, the noise power, N , con- 
tributed by each unit in the receiving system, and subsequently refer- 
red to the receiver-input terminals, is estimated in terms of the param- 
eters of each unit, such as its thermal temperature T , transmission 

x 

loss L , noise figure F , etc. In the following derivations it is 
xx 

assumed that non- reflecting impedance -matched conditions exist at the 
junctions of the various units in the receiving system, except (possibly) < 
at the receiver-input terminals, and that each unit is constructed with 
input and output positive -resistance terminations; hence each unit, 
preceding the receiver, delivers the "available" noise power, N , at 
its output terminals. See Appendix A for methods of estimating noise - 
power levels in the receiving system, including a discussion of the 
effects of impedance -mismatching at the receiver -input terminals. 



36. 



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-37- 
Ref erring to figure 4-1, the total receiving-antenna system output 

noise power, N' , available at the receiver-input terminals and within 

the bandwidth B (=B ) "accepted" by the receiver, is given by (A. 25) 

of Appendix A and is re-written here as (4.2) 



N' = kT. B. r (4.2) 

T A if v ' 

-23 
where, k = Boltzman's constant = 1.3804x 10 joules 

per degree Kelvin 

T = Effective output noise temperature of the receiv- 
ing-antenna system, degrees Kelvin 

B = Effective receiver IF bandwidth, in c/s, see 
1 (A. 18) 

The effective output noise temperature of the receiving- antenna 

system, T , is defined by (4.2). T is a function of the receiving- 

antenna system parameters and is given by (A. 30) of Appendix A, and 

reproduced here as (4. 3), 

* T a +( V 1)T c +L c (L t- 1) V L c L t (L du- 1)T du P H 

t a= rrr + S53i < 4 - 3) 

c t du if D 

The value of T , in degrees Kelvin, may be calculated by means 
of (4. 3), by substituting values for the various receiving-system param- 
eters; these parameters are completely described and defined in Ap- 
pendix A. 

If the thermal temperatures of the various units in the receiving- 
antenna system are equal to T , and if impedance -matching conditions 
exist throughout the receiving-antenna system, (4.3) becomes, 

T -T P.. 

a o n , a -, \ 

+ T + — — = — (4.3a) 



A L L Is, o kB.,L n 

c t du if D 

The portion of the noise power, N' , which is "delivered" to or 
absorbed by the receiver input circuit is, 



-38- 

N dT =M ir N T ' 4 - 4) 

where, M. = Impedance mismatch factor, at the receiver 

ir 

input terminals. This factor is a function of 

the impedance at the antenna- system output 
terminals, Z , and the receiver-input im- 
pedance, Z. , and is defined by (A. 16) as, 
ir 



4R R. 
M. = ^_il (A . l6 ) 

" l Z H +Z - I 
du ir 

where, R , R = Resistive or "real" component of the complex 

ir du 

impedances Z and Z , , respectively, 
ir du 

The noise power contributed by the linear section of the receiver, 
and delivered to the receiver-input terminals, is closely approximated by, 

M. {(F -l)kT B.J = M. kT B., (4.5) 

ir r oif ir erif 

where, F = Receiver noise figure [ IRE Comm. on noise, 

I960] , as a ratio 

T = Standard reference temperature, 290 degrees 
Kelvin [ IRE Comm. on noise, I960] 

T _ = Effective receiver input noise temperature [ IRE 
Comm. on noise, i960] , degrees Kelvin 

From (4. 5) we obtain: 

T = (F - 1)T (4.6) 

er r o 

The total noise power delivered to the receiver-input terminals 
is equal to the sum of the noise powers given by (4.4) and (4.5) or, 



N , + M. {(F -l)kT B.,} = M. (T A +T )kB.. (4.7) 

dT ir r o if ir A er if 



-39- 

The dependence of the total noise power at the receiver input, 

relative to the noise temperatures T . and T , is clearly evident from 

A er 

(4.7). 

The total predetection noise power level is equal to the total noise 

power' at the receiver input, M. (T . + T )kB. r , multiplied by the 
^ ^ lr A er if r y 

receiver gain, G . The load-reflected [ Siegman, 19&1] noise power 
in the predetection circuit of the receiver is assumed to be negligible 
for communication-type receivers, and is not considered here; how- 
ever, for extremely low-noise negative-resistance and/or bilateral 
types of receivers this noise power may be important. Hence, the 
total average noise power at the receiver-detector input point, N. ., 
is given by: 



N. r = G M. [T A + T ] kB. r (4.8) 

if r lr A er if 



where, G = Receiver power gain, as a ratio. This factor is 

defined as the ratio of the signal power delivered 
to the receiver input terminals to the available 
signal power in the receiver IF circuit. 

When using (4.8) to estimate the predetection total noise power 
level, N.,, the first step is to determine the numerical value of the 
factor T . From (4.3) it is seen that the numerical value of T 
depends upon the receiving-system parameters. All of these param- 
eters except T may be either measured or estimated with a high order 

of accuracy. T varies with time and antenna orientation and hence 

' a 

may (in some instances) be estimated only to a low order of accuracy. 

The antenna noise temperature, T , or noise power, N , depends 

a a 

upon the RF carrier- signal frequency [ CCIR Report 65, 1957] , 
[Crichlow, Smith, Morton and Corliss, 1955] , [Hogg and Mumford, 
I960] . However, in the microwave frequency range it may be assum- 
ed [ Grimm, 1959] that T varies between 50 to 300 degrees Kelvin, 

a 



-40- 

provided that the main lobe of the antenna pattern does not include the 

sun or an intense radio star. 

The remaining factors in (4.8), G , T , and B. r , are obtainable 
° r er if 

from the receiver specifications; the substitution of these factors in 

(4.6) and (4.8) together with the previously calculated Value of T , 

gives the predetection noise power level, N. f , and the receiver-input 

total delivered noise power level, M. (T . + T )kB.., for the 

lr A er if 

estimated and the specified receiving- system parameters. 



-41- 

4. 3 Post-Detection Noise Power Levels 
in the Receiving System 

Post- detection noise, which appears at the receiver output, is 
generated at various points within the system and is due to various 
causes as listed below: 

(1) Pre -detection (amplitude) noise, N , at the detector input. 

(2) Noise power at the modulator input, N , relative to the 

m 

total modulating- signal power, P 

m 

(3) Inter modulation noise, generated within the equipment. 

(4) Radio-wave path-modulation noise, imposed on the RF signal 
by transmission over the radio-wave path. 

A complete analysis of post-detection noise requires a considera- 
tion of the statistical properties of each of the above noise powers in 
order to obtain the characteristics of the resultant total noise power 
per message -signal channel, N , at the receiver output. The effects 

of N on system performance may then be determined in terms of 
oc 

the resultant average signal power -to -average noise power ratio, 

S /N , at the receiver output, 
oc oc 

Throughout this analysis it is understood that the post-detection 
noise is "available" at the point where it is being measured; usually, 
noise -power measurements are made for impedance -matched condi- 
tions and within a particular and carefully defined effective bandwidth. 
In FM systems, the post- detection noise level will depend upon its 
position in the baseband- signal spectrum, and will be a maximum in 
the top channels of the baseband signal, when the receiver -input 
signal is above the threshold point. If noise power levels are meas- 
sured at a point beyond the receiver output, such as the output termi- 
nals of a multiplex unit, due allowances must be made for the noise 
properties of the intervening units and the power gains or losses in 
these units . 



■ 42- 



FM RECEIVER-OUTPUT SIGNAL AND NOISE IN 
THE TOP CHANNELS OF THE BASEBAND SIGNAL 



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Pr 
RECEIVER -INPUT RF-SIGNAL AVERAGE POWER 



Figure 4 -A 



-43- 

4.3. 1 Post-Detection Noise Due to Pre-Detection 
Amplitude Noise 

The effect of pre -detection noise, item (1) above, on the post- 
detection noise level, shown by the N curve in figure 4- A, is con- 
sidered in detail in sections 4. 5 and 4. 6. The design equations and 
the design curves developed in this report are based primarily on the 
effects of the pre -detection noise with the understanding that the 

transmitter -output (amplitude) noise, N , was not included in the 

at 

pre -detection noise. The reason for this omission may be seen from 
an inspection of figure 4-A, where the N curve gives the receiver- 
output noise, which is due to the pre -detection noise, N , excluding 
the transmitter -output noise. The receiver -output noise which is 
due to the transmitter -output noise is dependent upon the transmitter - 
output signal-to-noise ratio and is shown by the N curve. It is 
evident that the receiver -output noise, N , effectively limits the 

maximum value of the receiver -output signal- to-noise ratio, S /N 

r b oc oc 

Modifications of the receiver-performance characteristics for 
the effects of that portion of the post-detection noise which is due to 
items (2), (3), and (4), are considered below. 

4.3.2 Post-Detection Noise Due to Noise 
at the Modulator Input 

The design equations and curves in this report were based on the 
assumption that the signal-to-noise ratio in the modulating (baseband) 
signal, at the modulator input, was very much greater than the required 
(baseband) signal-to-noise ratio at the radio receiver output terminals. 
The net effect of noise at the modulator input is to place an upper limit 
to the receiver -output signal-to-noise ratio. In general, we cannot 
expect to obtain a signal-to-noise ratio at the receiver output higher 
than the signal-to-noise ratio at the modulator input. The post-detection 



-44- 
or receiver-output noise -power level, due to noise N , at the modu- 
lator input, is shown by the N curve in figure 4-A. 

Estimated and measured system-performance characteristics must 
involve the baseband signal-to-noise ratio at the modulator input, as a 
factor which limits the radio receiver -output signal-to-noise ratio. 
The actual value of the receiver -output baseband signal-to-noise ratio 
will be lower than the modulator -input baseband signal-to-noise ratio, 
and will depend upon the signal-to-noise "degradation factor" of the 
radio -frequency section of the communication system. 

4.3.3 Equipment-Intermodulation Post-Detection Noise 

Inter modulation noise is generated within the equipment whenever 

the limits of linear modulation or demodulation are exceeded and when 

the receiver IF effective bandwidth, B._, is insufficient. For FM 

if 

systems, the post-detection noise power level, due to equipment 
intermodulation noise, is shown in figure 4-A, by the N curve. 
Hence, the post-detection (equipment) intermodulation noise-power 
levels may be estimated from the characteristics of the modulator, 
the demodulator, and the passband characteristics of the receiver IF 
circuits, using the material in Section 3. 

For amplitude modulation, the required modulator characteristic 
is in terms of the sinusoidal modulating- signal power level, P , 
versus the percent modulation. The percent of time during which 
100% amplitude modulation is exceeded, for particular power-levels 
of a noise -type modulating signal, P , may be obtained from figure 
3-5. The percent of time during which 100% amplitude modulation is 
exceeded may be used as an estimate of the percent of the modulating 
signal power which is converted to noise power. By arbitrarily assign- 
ing a limit to the percent of time during which 100% amplitude modulation 



-45- 
may be exceeded, the value for D __ may be obtained from figure 3-5. 
At the same time, the corresponding value for the average power level 

of the noise -type modulating signal, P , may be obtained from 

mn 

figure 3-5. It is evident from figure 3-5 that intermodulation noise 

in the modulator is dependent upon the factor D ; low values of inter - 

ss 

modulation noise require high values for D , and vice versa. 

ss 

If the amplitude demodulator is linear to 1 00% amplitude modula- 
tion of the RF signal, it will not add appreciable intermodulation noise 
power to the system. 

For frequency modulation, the required modulator characteristic 

is in terms of the sinusoidal modulating- signal power level, P , 

& B ^ ms 

versus the peak frequency deviation, AF . The modulator character- 

s 

istic should be linear for values of A F somewhat greater than AF 

° s 

in order to prevent excessive "distortion" of the RF spectrum during 

the time when the peaks of the noise -modulating signal occur. This 

distortion is comparable to that produced in an amplitude modulator 

when the modulating- signal power level exceeds P (100%); the net 

° ° ms 

effect results in distortion of the baseband signal at the demodulator 

output, and the production of receiver -output noise, shown by the N . 

curve . 

The relationship between P /P , for FM systems, and the 

ms mn 

required RF spectrum bandwidth, B f , is shown in figure 3-4. The 

effect of "bandwidth clipping" in the receiver, on the receiver -output 

signal-to-noise ratio, S /N , is shown in figure 3-2. By combining 

oc oc 

the modulator characteristic with the data in figures 3-2 and 3-4, it is 
possible to estimate the relationship between the increase of the post- 
detection noise level or the degradation of the receiver-output 



-46- 

signal-to-noise ratio, and the modulating- signal power level, P 
o ° mn 

The results will be relative to the FM receiver threshold point and 

the bandwidth ratio, B r /B... 

rf if 

If the FM demodulator characteristic is linear to ± A F it will 

n 

not add an appreciable amount of intermodulation noise; however, if 
this condition does not hold the demodulator characteristics 
should be included in the above analysis of intermodulation noise 
effects . 

The portion of the receiver-output noise power, N , due to 
modulator-transmitter -receiver intermodulation effects, may also be 
measured by means of a transmitter-receiver back-to-back test. For 
this test, the receiver performance characteristic is first measured 
using a modulated RF signal having a negligible amount of intermodu- 
lation noise; this condition is attained by using a comparatively low 
value for the modulating -signal power level, P . For multichannel 
systems, the modulating signal should be white noise which is trans- 
mitted through a band-rejection slot filter; noise and signal measure- 
ments are made at the receiver output within the clear channels or 
spectrum "slots." Meaningful results will only be obtained if the 
entire system is properly adjusted for normal operation. The modu- 
lating- signal power level, P , is then increased in steps, and either 
or both the receiver -output increase in noise or the change in the 
signal-to-noise ratio are measured for each modulating-signal power 
level, P . , and for a range of receiver -input RF signal power levels, 

P . From the results of these tests it is possible to determine the 

r r 

relationship between the modulating-signal power level, P , and the 
amount of intermodulation-noise power; in terms of the received RF 
signal power, P , required to produce a given receiver -output signal- 
to-noise ratio--for particular values of the modulating-signal power, 



-47- 

P . For FM systems the results would be similar to those shown in 
m 

figure 4-A. 

Since the design curves in this report were measured and calculated 
under conditions of negligible intermodulation noise power; the above 
information may be used to make the proper corrections for the effects 
of equipment intermodulation noise in the design curves; see section 
4.3.6. Alternatively, departures of system-performance measured 
results from design-curve values may be attributed in part to equip- 
ment intermodulation-noise effects, and corrections may be made 
from the above data, after due allowances have been made for all other 
factors. 

The results of the above intermodulation-noise tests will include 
intermodulation effects in both the modulator -transmitter unit and the 
receiver. Furthermore, the relationship between the modulating- 
signal power level, P , and the system intermodulation noise power 
will be obtained; these equipment-performance data should be used as 
guides for setting up system-performance tests and for operation of 
the particular system under test. The above types of tests may also 
be conducted on a complete system, utilizing an over -the -path signal. 
In fact, this method of testing is necessary in order to determine the 
performance characteristics of the combiner and to obtain measure- 
ments of the path -modulation noise; see section 4.3.4. A complete- 
system test is not as convenient to conduct as an equipment back-to- 
back test, primarily because of the lack of control of the received RF 
signal, in the former case. 

4.3.4 Post-Detection Path- Modulation Noise 

Radio wave path -modulation effects on the transmitted RF signal 
are due to "frequency selective fading" of the RF-signal spectrum 



-48- 
components, which results in distortion of the received-signal spectrum 
and consequent distortion of the baseband- signal components. Path 
modulation arises primarily because of the differences in the relative 
delays of the (received) RF- signal spectrum components due to multi- 
path conditions. In a multichannel FM system, the effects of path 
modulation appear at the receiver output as interchannel modulation 
noise. In an AM system the effect of path modulation on the receiver- 
output noise is somewhat different from that in an FM system; however, 
the overall effects on system performance are similar for both types 
of systems. 

The severity of selective fading increases with the RF-spectrum 
bandwidth and hence determines the "transmission bandwidth capability" 
of a radio-wave path. Selective fading, and therefore, transmission 
bandwidth capability, depend also on the type of RF signal, such as 
HF via ionospheric reflection, tropospheric, line of sight, etc. 

A measure of the degree or the amount of selective fading may be 
obtained from the cross -correlation of the amplitudes of the spectral 
components of the received RF-signal. However, a more direct and 
useful measure of the transmission-bandwidth of a radio-wave path 
may be obtained by measuring the path -modulation noise in the base- 
band signal at the receiver output [Clutts, C. E. , et al, I960] . The 
power level of the path -modulation noise will depend upon the RF- 
signal spectrum bandwidth, B , and the various system-path para- 
meters, such as path length, antenna beamwidth, and to some extent, 
the order of diversity and the RF-signal frequency. This noise is 
time -varying and hence its study requires statistical treatment. 

Path -modulation acts the same as signal modulation and sets a 
lov/er limit to the receiver -output noise and hence, determines an 
upper limit to the receiver -output signal-to-noise ratio, regardless 



-49- 

of the power transmitted or received. For a given system, RF band- 
width, and type of RF signal, any apparent dependency of the median 
receiver -output path -modulation noise level on the median received 
RF signal level is due primarily to variations of the type of received 
RF signal. However, the path -modulation noise level at the receiver- 
output will also appear to be dependent upon the median received RF 
signal, for RF- signal levels below the receiver threshold point; see 
figure 4-A. Hence, the net effect of path -modulation noise is similar 
to the effects of either noise at the modulator input, or equipment 
inter modulation noise. Receiver -output noise power, due to path- 
modulation effects, is shown by the N curve in figure 4-A. 

4. 3. 5 Total Post-Detection Noise 

The total receiver -output noise -power per message -signal channel, 

relative to the received RF- signal, P , is shown by the N curve in 

r oc 

figure 4-A. The absolute levels of the noise-power curves, N to 

N , inclusive, are dependent upon the various factors described 
05 

above. These curves are positioned as shown in order to indicate 

their relative effect on the level of the noise curve, N , and the 

oc 

resultant effect on the receiver -output signal -to -noise ratio curve, 

S /N . The shapes of the curves are dependent upon the type of 
oc oc 

modulation-demodulation employed and include the effects of age 
(automatic gain control) and limiting, in the receiver. For the case 
where system back-to-back and over-the-path tests are made, the 
curves in figure 4-A may be measured accurately. However, it 
should be understood that the absolute levels of the post-detection or 
receiver-output noise powers, N to N , inclusive, are dependent 
upon power gains (or power losses) within the receiver, therefore, 
these noise-power levels are meaningless unless they are referred 



-50- 

to the post-detection or receiver -output signal - power level, S 
Hence, estimates and measurements of receiver -output noise power 
levels should only be made for conditions where signal-power levels 
are properly adjusted throughout the system. 

4. 3. 6 Effect of Post-Detection Noise on Receiver 

Characteristics and Method of 

Correcting System-Design Curves 

The actual performance characteristic of the receiver will be as 

indicated by the "corrected" receiver -output signal-to-noise ratio 

curve, (S /N )', in figure 4-A. The corrected curve is obtained 
oc oc 

from the "ideal" curve by reducing the ideal-curve values of S /N 

y B oc' oc 

an amount equal to the noise -power-level increase (in db units) between 

the 1ST . curve and the N curve; for particular values of P . These 
01 oc r 

corrections also apply to the design curves in this report, which are 
identical to the ideal curve shown in figure 4-A. In practice, correc- 
tions to the design curves would be in terms of the additional required 
receiver -input RF-signal power, AP . This additional power would 
be required to attain a particular value of receiver-output signal-to- 
noise ratio, (S /N ) . , or a particular grade of service. The actual 
oc' oc i r 6 

value of AP would be approximately equal to the noise -power -level 

difference between the N . curve and the N curve. Note that this 

01 oc 

type of correction is limited by the "levelling -off " point of the corrected 

receiver-characteristic curve; beyond this point, no improvement in 

the receiver -output signal-to-noise ratio will be obtained by increasing 

the receiver-input RF signal power, P . 

r 

The net effect of post-detection noise on the performance of a radio 
communication system may be determined from the corrected receiver 
characteristic curve. The use of the corrected receiver characteristic 
curve results in an accurate determination of the distribution of the 



-51- 

time-varying post-detection (baseband) signal, noise and signal-to- 
noise ratio; these distributions are very important factors in the per- 
formance of either single-link or tandem-link radio systems. The 
performance of single-link and tandem-link radio systems should be 
determined from the distribution of the time-varying post-detection 
(combined) baseband signal-to-noise ratio. Any attempt to estimate 
the performance of radio systems solely in terms of the pre-detection 
signal-to-noise ratio, is likely to lead to the erroneous conclusion that 
this type of solution is complete and general. 

For single-link systems, the characteristics of the post-detection 
or receiver-output signal, noise, and the signal-to-noise ratio would 
be obtained by the methods outlined above. 

In tandem-link systems, the signal and the noise at the intermed- 
iate points in the system may be "transferred" either at the pre-detect- 
ion point or at the post-detection point. If the signal and the noise are 
transferred at the pre-detection point, signal de-modulation is not in- 
volved at the intermediate points in the system and the receivers are 
used as linear repeaters at these points. Hence, in this type of system 
the characteristics of the intermediate -point pre-detection combined 
signal-to-noise ratio will influence the transmitted and the received 
(RF) signal-to-noise ratio. However, in a linear-repeater type of 
system, the system-intermodulation noise which is developed within 
each of the tandem links, is transferred through the system and 
arrives at the output of the terminal demodulator. Therefore, in a 
tandem-link system which employs linear repeaters at the intermediate 
points, the total system-intermodulation noise at the terminal-demodu- 
lator output may be approximated by the sum of the system-intermodu- 
lation post-detection noise in each of the tandem links. 



-52- 
In tandem-link systems employing demodulation at the intermed- 
iate points in the system, the characteristics of the post-detection or 
receiver-output signal and noise at each intermediate point would be 
determined by the methods outlined above for single-link systems. 

The total receiver-output noise, N , at an intermediate point in the 

oc 

system would be applied to the input terminals of a modulator, and 

hence this noise power (N , see section 4. 3. 2) would influence the 

m 

system-intermodulation noise developed in the succeeding link of the 
system. Under these conditions, the performance of each succeeding 
link in a tandem-link system becomes progressively worse because of 
the progressive deterioration of the signal-to -noise ratio at the modu- 
lator input terminals. 

Therefore, the characteristics of the post-detection noise at the 
output terminals of a tandem- link system will depend somewhat upon 
whether the intermediate links are operated as 

(a) linear repeaters, which transfer the received signal and 
noise at each intermediate point in the system to another 
portion of the RF spectrum; or 

(b) as receiver-demodulator units which produce the baseband 
noise (and signal) at each receiving-point in the system. 

In either case, the characteristics of the post detection noise may be 
determined by the methods outlined in section 4. 3. 

The case of re-construction of the message signal at each inter- 
mediate point in a tandem-link system is equivalent to single-link 
operation, and the total message errors would be equal to the sum of 
the message errors in the individual links. 



-53- 
4.4 System Parameters Which Affect the Message-Channel 
Signal-to-Noise Ratio at the Radio Receiver Output 

In the previous Subsections we discussed the effects of predetect- 

ion and post-detection noise on the receiver -output signal-to-noise 

ratio, S /N . In this Subsection we will consider the relationship 
oc oc r 

between the message -channel signal-to-noise ratio, S /N , at the 

oc oc 

radio receiver output, and the following factors: (1) the average total 
power of the composite modulating signal, P , (2) the individual mes- 
sage-channel average signal-power level, p , in the modulating base- 
band signal, relative to the total average power level of the modulat- 
ing baseband signal, P , and (3) the message-signal spectrum band- 
width, B . It is shown later that these parameters determine the per 
c 

message -channel signal-to-noise ratio at the radio receiver output, 

in terms of the total RF signal power-to-total noise power ratio and 

the RF spectrum bandwidth, B ., at the receiver input. 

It should be noted that the message-channel signal-to-noise 

ratio, S /N , at the message-signal decoder-unit input, is related 

to the radio receiver output total signal-to-noise ratio, S /N , through 

the characteristics of the receive-multiplex equipment. The net result 

of this situation is that for some types of message -signal multiplexing 

equipment [ Landon, 1948] , the message-channel signal-to-noise ratio, 

S /N , cannot be measured directly in the radio receiver output base- 
oc oc 

band signal. For SSB frequency-division multiplex, being considered 

in this paper, S /N may be measured directly in the receiver output 
oc oc 

baseband signal, in terms of the message-signal spectrum bandwidth 

and its position in the baseband signal. For example, in FM multiplex, 

S /N cannot be measured directly in the radio receiver-output base- 
oc oc 

band signal, and must be determined at the output terminals of the 
receive-multiplex equipment. 



-54- 

Hence the multiplex receiver characteristics must be included 

in the analysis, in terms of the total baseband signal-to-noise ratio, 

S /N , at the multiplex receiver input, and the message -channel 
o o 

signal-to-noise ratio, S /N , at the multiplex receiver output. The 

oc oc 

above considerations make proper allowances for any change in the 
message -channel signal-to-noise ratio which might be attributed to 
the message- signal multiplexing system. 

4.4.1 Dependence of Receiver Output Signal-to-Noise Ratio 

on the Modulating -Signal Power Level, P 

m 

In SSBSC-AM systems the power level of the modulating signal, 

P , affects the receiver output signal-to-noise ratio only in terms 

of the intermodulation noise generated within the modulator. The 

dependence of the receiver output signal-to-noise ratio on the average 

power level, P , of the noise-type modulating signal is referred to 

the power level, P ^ (100%) of a sinusoidal modulating signal; where 

P ^ (100%) is the modulating- signal power level required to provide 

100% a.mplitude- modulation of the RF carrier signal. The factor, 

P /P (100%), may be obtained from figure 3-5. 
mn' ms * ' ' B 

Note that for a sinusoidal modulating signal, P or, P (100%) 

m ms 

for SSBSC-AM systems and P for FM systems, would be obtained 

ms ' 

from the modulator characteristic. For a noise-type modulating 

signal, P or, P would be obtained from figure 3-4 for FM systems, 

m mn ° ' 

and from figure 3-5 for SSBSC-AM systems. 

In FM systems, the modulating-signal power level, P , deter- 
mines the required RF- signal spectrum bandwidth, B . The relation- 

rf 

ship between the factor, P /p , and B ,. may be obtained from 

mn ms rf 



figure 3-4. Note that for the FM case, the reference power level is 

P and is not a li 
ms 

SSBSC-AM system. 



P and is not a limiting factor as is P (1 00%), which is used in the 
ms ms 



-55- 

A value for B . is required in order to establish the noise-power 

rf c 

level at the radio receiver input, see Section 4. 2. The radio receiver 

IF bandwidth, B.., is associated with B „, and hence these factors 
if rf 

influence the radio receiver output signal-to-noise ratio. The re- 
lationship between the receiver output signal-to-noise ratio, S /N , 

o o 

and B , for FM systems, is derived fully in Sections 3.2, 4.6, and 
Appendix C. In SSBSC-AM systems, the radio-frequency signal spec- 
trum bandwidth is independent of the modulation index, and is approxi- 
mately equal to the highest frequency in the modulating baseband signal. 
The portion of the radio-receiver output noise power which is re- 
lated to the modulating signal power, P , is known as "intermodulation 
noise". This intermodulation noise is generated in a radio communi- 
cation system whenever the limits of linear modulation or demodu- 
lation are exceeded, in either or both the radio transmitter and the 
radio receiver. It is difficult to estimate intermodulation noise power 
levels and accurate results must be based on the equipment perform- 
ance; however, it is usually assumed that the intermodulation noise 
power level is equal to the thermal noise power level, for practical 
system design. See section 4. 3 on equipment intermodulation effects. 



-56- 

4.4.2 Dependence of Receiver Output Message-Channel Signal-to-Noise 

Ratio, S /N , on the Message -Channel Signal- Power Loading 
oc oc 

Ratio, P /p , in the Modulating Baseband Signal. 

The message-signal power, p ., which must be allocated to each 
individual message signal, in the modulating baseband signal, depends 
upon the required signal-to-noise ratio at the radio receiver output. 
This required radio-receiver output signal-to-noise ratio depends upon 
the specified grade of service or message error rate, for a particular 
message. The relationship between the radio receiver output message- 
channel signal-to-noise ratio and the message error rate is discussed 
in Section 5. 

In this work it is assumed that the individual message -signal 
average power level in the modulating baseband signal, must be ad- 
justed so as to give a specified or required average message-channel 
signal-to-noise ratio at the receiver output. Under these conditions 
(see Appendix B), the message-channel signal-power loading ratio 
in the modulating baseband signal is: 



M 



m 



oc 



cV N 



oc 



+ . . + M 



oc 



cV N 



+ . . +M 



oc/ J I 



N 



B 



oc 



cV N 



oc 



mi 



B 



ci 



S N 
oc ^ 

N~~ J. 
oc A 



X 



(4.9) 



-57- 
where, p . = Average power level of a particular message 



mi 



signal in the modulating baseband signal. 



P = Average total power level of the composite 

modulating baseband signal, under conditions 
when a specified number of simultaneously- 
active message channels are operating. For 

sinusoidal- signal modulation, P = P , and 
j- • ■ -, ■. _ m _ ms 

for noise -signal modulation, P = P 

m mn 

B . = A particular message -signal channel bandwidth 

in c/s, at the receiver output. This channel 

bandwidth is assumed to be equal to the spectrum 

bandwidth of the message signal. 

S = A particular message-channel signal average 

power level, at the radio receiver output, or a 
related message-channel signal average power 
level at the decoder -unit input. 

N __ = Average noise-power level of a particular 

mess age -channel having a bandwidth B , at the 
radio receiver output or, a related noise-signal 
average power level at the decoder -unit input. 

S N 

oc \ 
— — ) = Required radio receiver output signal-to-noise 

oc/ i ratio, or required decoder-unit input signal-to- 
noise ratio plus a degradation factor for the 
multiplex receiver, in a particular message- 
signal channel, for a specified allowable average 
message error rate. 

M = Number of simultaneously-active message sig- 
nals each of which requires the same value for 

the product {B (S /N )} A . 
c oc oc A 

M T = Ditto, for the product {B (S /N )} t 
I c oc oc I 

M„ T = Ditto, for the product {B (S /N )>. 
N ^ c oc 7 oc N 

As noted in Subsection 4. 3, for some types of message-multiplex- 
ing systems, it may be necessary to determine the message-channel 
signal-to-noise ratio at the output of the multiplex receiver because of 
the impossibility of measuring this ratio directly in the receiver-out- 
put baseband signal. Conversely, at the transmitting end of the system 



-58- 
for some types of message-signal multiplexing, it may not be possible 
to measure the modulating signal power levels directly in the modu- 
lating baseband signal: under these conditions, p and P must be 
measured at the multiplex transmitter input. This situation will not 
arise where SSB frequency-division multiplex equipment is employed. 
In order to use (4.9) it is necessary to determine the required 

message-channel signal-to-noise ratio, S /N , and the message- 
6 & oc / oc B 

signal bandwidth B , at the radio receiver output. This required infor- 
mation would be obtained from measured or estimated data such as 
shown in figure 5-3 of Section 5. 

The specified message error rate depends upon the type of mes- 
sage. Consequently, the required receiver -output signal-to-noise 
ratio will be different for each type of message. However, if different 
types of messages require the same value for the product, \B (S /N )} , 
then these messages may be grouped in the same class, M . 

Proper methods for adjusting the power levels of the individual mes- 
sage signals, at the terminals of the multiplexing equipment, are out- 
lined in the manuals for this equipment; the instructions in these man- 
uals should be followed carefully in order to avoid overloading of the 
multiplex circuits, and thereby prevent consequent intermodulation 

distortion. The power levels p . and P must be considered in terms 

mi m 

of their effect at the radio-transmitter modulator input; if the multiplex 
baseband power level is inadequate, a power amplifier or an attenuator 
is required at this point in the system. 



-59- 

4. 5 Performance Characteristics of SSBSC-AM Radio Receivers 

In this section the performance of SSBSC-AM (single sideband 

suppressed carrier amplitude modulation) radio receivers is defined 

in terms of the signal-to-noise ratio per message channel at the 

receiver output, S /N , versus: (a) the total received RF signal 

oc oc & 

power and (b) the total available noise power at the radio receiver 
"front-end" input. In order to obtain accurate estimates of the radio 
receiver performance it is necessary to consider the statistical 
characteristics of the received RF signal power and the noise power, 
at both the receiver input and at the receiver output. Methods for 
estimating the noise level and the characteristics of the noise at the 
receiver input are outlined in Section 4. 2 and Appendix A. It is 
assumed here that the total noise -power at the receiver input approxi- 
mates "white " noise and that the power level of this total noise is pro- 
portional to the RF spectrum bandwidth accepted by the receiver. 

The effects of system-intermodulation noise are not included in 
the analysis of receiver performance, at this point, and hence, the 
results will be in terms of "ideal" receiver performance. Modifi- 
cations required for the effects of system-intermodulation noise on 
the ideal receiver-performance characteristics, are outlined in 
section 4. 3. These remarks apply also to FM-receiver performance, 
as developed in section 4.6. 



-60- 

Since SSBSC-AM receivers are linear, it follows that the character- 
istics of the receiver -output noise are similar to the characteristics 
of the noise at the receiver input, within the limits of receiver linear- 
ity. 

The ideal SSBSC-AM receiver does not exhibit a "threshold effect", 
that is, there is a linear relationship between the receiver-input signal 
power level and the receiver -output signal-to-noise ratio. The follow- 
ing procedures for determining SSBSC-AM receiver performance are 
also applicable in case the receiver characteristic curve is non- 
linear, and is known. 

In this work the relationship between the receiver output signal- 
to-noise ratio and the receiver input-signal power level was calculated 
for SSBSC receivers, assuming that the receiver-input RF signal was 
steady, that is non-varying with time. To obtain the receiver-perform- 
ance characteristic for a time-varying receiver-input RF signal, the 
above steady receiver-input signal receiver -output signal-to-noise 
ratio characteristic was combined with the distribution of the time- 
varying receiver-input signal; see Appendix D. 



-61- 
4.5. 1 Receiver -Input Radio-Frequency Signal Power Requirements 

for SSBSC-AM Systems Using Steady Received Signals. 
For an ideal SSBSC-AM receiver we have, 



— = — X — (4.10) 

o if 



where, P = Total of the average IF signal power at the 

receiver demodulator input, in watts. 

N. f = Total of the average noise power at the receiver- 
demodulator input, for IF bandwidth B. r , in watts 

if 

S = Total of the average baseband signal power at 
the demodulator output, in watts 

N = Total of the average noise power at the demodu- 
lator output for baseband bandwidth ~B... 

On a per message-channel basis, with a message-channel signal- 
er loading ratio of P /p in the modulatin 
5 iri *m 

radio transmitter (see Section 4.4. 2) we have, 



power loading ratio of P /p in the modulating baseband signal at the 



S S p B p 

-2£ = -° x -^ x _Jt! ( 4.ii) 

N N P B V 

oc o m c 



or, from (4. 10) and (4. 11) 



S p B - f P. f 

oc m if it 



x 



N P B N 

oc m c 



x tt 1 - (4.12) 



if 



where, S = Signal power in the receiver output, per mes- 

sage-signal channel, in watts. 

N = Noise power in the receiver output in a message- 
oc 

signal bandwidth, B , in watts. 

p = Average power level of a message-signal in the 



m 



modulating baseband signal, in watts, 



-62- 

P = Total average power level of the composite 

modulating baseband signal; determined under 
conditions when a specified number of simul- 
taneously-active message signals are operating, 
in watts. 

B = Message- signal bandwidth, in c/s. 

B. = B = Effective bandwidth at the radio receiver IF 
if rf 

output, or demodulator input. 

In an SSBSC-AM system the radio-frequency spectrum bandwidth 
B , and also the receiver IF bandwidth, B , are approximately equal 
to the highest frequency, f , in the modulating baseband signal; hence, 

B = B. * f (4.13) 

rf if m 

Combining (4. 12) and (4.13) and rearranging terms, we obtain, 

S B P P., 

oc c m it . . , . . 

ST" x - x — = n~ (4 - 14) 

oc m m if 

The term, P /N , in (4.14) is the required predetection signal- 
to-noise ratio, associated with a specified value of S /N at the 

oc oc 

receiver output, for particular values of the parameters: f , B , 

m c 

and P /p . However, in radio communication system design and 
mm 

test work it is much more convenient to deal with the radio-frequency 

signal power and the noise power at the radio receiver input terminals 

than it is to measure P._/N._ in the IF section of the receiver. Hence 

if if 

the form of (4.14) was modified in the following derivations to include 

the total received signal power, P , and total received noise power. 

r 

First, 

P.. = G M. P (4.15) 

if r lr r 

P P 

*The symbol R is sometimes used to designate — - — or 10 Log 



N. r 6 10 N. f 

if if 



and (see 4. 8), 



-63- 



N = G M. {T A + T }kB. r 
it r lr A er if 



or, since B. . « f for the SSBSC-AM case, 
if m 



N. f = G M. {T A +T }kf (4.16) 

if rirAerm 



therefore, 



P., P 

if r 



N., k(T A + T ) f 

if A er m 



(4.17) 



where, G = Power gain of the cascaded RF and IF linear 

r 

sections of the SSBSC radio receiver 

P = Total average radio-frequency signal power 

available at the receiver input, in watts 

-23 
k = Boltzman's constant = 1.38 04 x 10 joules 

per degree Kelvin 

M. = Defined by (A.16); see Appendix A. 

f = Highest frequency in the modulating signal 

T = Effective receiving-antenna system output noise 
temperature, degrees Kelvin. See Section 4. 2 
and Appendix A on noise at receiver input. 

T = Effective receiver-input noise temperature, 

degrees Kelvin [ IRE Comm. on Noise, I960] . 

Substituting (4. 17) in (4. 14) we obtain, 



S B P P 
oc c m r IA ... 

— X T— X — = k(T A+ T )f (4 ' 18) 

oc m m A er m 

The factor f was retained in both sides of (4. 18) in order to 
m 

obtain power-level values of P on the abscissa scale of the design 

r 

curves shown in figure 4-2, relative to the total noise power, 

k (T + T )f , at the receiver input. 
A er m 



64 



SSBSC-AM RADIO RECEIVER PERFORMANCE CURVES 
STEADY AND RAYLEIGH- FADING TYPES OF RECEIVED SIGNAL 



o_ e L E 



- 35 



Q 































i 1 ! ! ! 


' 




' 


, 


' ' ' ' 1 / y 
















































































y s 










































































































. "h_.J_!_Ll 


































-f MAXIMAL-RATIO COMBINING - 






























































1 I 


. 


































\ 


1 






S s r X 




































• ' XX XX 




























III 












































































































































X^ i/^ rS 




































f s 




































sS 
























± 




















































jf 


















































































































r r r r 




























































































































, | 




















































































































































































































































































QUADRUPLE C 


-iJ i XX 






















































































































II III 








































. 










































i DUAL on 


/bHSIIY-^ > 








































1 


1 ^> 








































Lf 
















































































JT 'S 








































jr * 


rJr-— 


r NON DIVERS 


TY 






























± 




* * 


Jr 






































n I 








































III ! 












































^1 i 












































~^-ST 


EADY S 


IA1 


















































































III M 


I 




































! ! ! Mi: 






































































/ 


i 1 
















1 














r Jr J* J 




! : i . . 
































































































































































nil 


































r r 






































































f 


y !/T 


* 




































& j» 




1 ' ■ 














































































>T r 
















































































f 






















































































































....:.... 




























































































































































^ 








































































































! ' ' 


1 
































1 / 


















1 




' 


























Jr 


















' ' 
















































































































































































, . 




















































































^_ 




























































































































































































































































1 










































II 





















































-5 



-15 

-240 -235 -230 -225 -220 -215 -210 -205 -200 -195 -190 -185 -180 

10 L06 o P. -10 LOG c f m -10 UX3 C <W> 
AVERAGE OF THE TOTAL SIGNAL POWER , WATTS AJ receiver wput 



J: 



-?: 



/STEADYN P, 

VSIGNAL^ WW AVERAGE OF THE TOTAL RELATIVE NOISE POWER 



/TIME- VARYING 
\ SIGNAL 






SAMPLING -PERIOD MEDIAN OF THE NON- DIVERSITY 

AVERAGE TOTAL SIGNAL POWER, WATTS 

")" SAMPLING -PERIOD AVERAGE OF THE TOTAL RELATIVE NOSE POWER 

Figure 4-2 



a *£:£.£= N«n 



-65- 
Equation (4.18) may be used to calculate the required radio-fre- 
quency signal power level, P , at the (SSBSC-AM) radio receiver input, 

to yield a specified receiver-output signal-to-noise ratio, S /N 

oc oc 

-for particular values of the parameters: f , P /p , B , T , and 

m m m c A 

T . However, it is more convenient to use the information con- 
er 

tained in this equation in a graphical form. Hence, (4.18) was plotted 

as shown in figure 4-2 for a time -invariant or steady RF signal at the 

receiver input. Scale values in figure 4-2 are normalized in terms of 

the parameters in (4.18). 

To obtain quantitative values for P from figure 4-2, correct 

values must be used for S /N , f , and p /P . The required 

oc oc m mm 

value of S /N is obtained from data similar to those shown in 
oc oc 

figure 5-3. f is taken to be equal to the highest frequency in the modu- 
lating baseband signal, in c/s. The message-channel signal-power 
loading ratio, P /p , may be calculated by means of (4.9). The 
factor T may be calculated from (4. 3). The radio-receiver input 
noise temperature, T , depends largely upon the carrier (radio) 
frequency and the type of receiver; T may be calculated by means of 
(4.6). Procedures for using the above information to estimate the 
performance of SSBSC-AM systems are outlined in Section 6. 

P if 

The predetection signal-to-noise ratio, — — , which corresponds 

if 
to a particular value of P , may be obtained from (4.17). It should be 

noted that (4.18) involves the modulation index, m , which is tacitly 

assumed to be unity, for SSBSC-AM systems, that is, P (100%) is 

7 J ms 

used as the reference level of the modulating- signal power. If the 
modulation index exceeds unity the resultant (intermodulation) distor- 
tion will appear as additional noise at the radio receiver input, and at 
the receiver output and will degrade the signal-to-noise ratio at the 
receiver output, for all values of receiver input power, P . 



-66- 
4.5 .2 Receiver -Input Radio-Frequency Signal Power Requirements for 
SSBSC-AM Systems Using a Rayleigh-Fading Type of Received Signal. 

The value of the required receiver-output signal-to-noise ratio, 

S /N , will not be the same for time-varying received RF-signal 
oc oc 

amplitudes as for steady received signals for the same message error 

rate. Also, S /N will depend upon the order of diversity. The 
oc oc 

value of the ratio, S /N , to be inserted in (4. 9) may be obtained 

oc oc 

from either steady-signal or varying -signal data similar to those 
shown in figure 5-3. However, each different type of message will 
require a different set of curves similar to those shown in figure 5-3. 

The curves in figure 4-2 for non-diversity, dual diversity and 
quadruple diversity were obtained by combining the SSBSC-AM steady- 
signal characteristic curve (in figure 4-2) with the (maximal- ratio) 
combiner characteristics given in figure D-lb, Appendix D. Details 
of the method used to combine these characteristic curves are given 
in Appendix D. It should be noted that the abscissa-scale values in 
figure 4-2 for the fading-type signals refer to the median of the non- 
diversity received total RF signal average power, P , while the 
ordinate- scale values yield the average of the time-varying signal-to- 
noise ratio, S /N , at the receiver output; all other parameters are 

OC OC f > f 

the same as for the "steady-signal" case. The 1.59 db displacement 

between the "steady-signal" curve and the "non-diversity" curve in 

figure 4-2 is due to the fact that the average value of the Rayleigh- 

distributed received RF signal power, P , is (1.59 + 10 Log n)db 

r 10 

greater than the median value of P ; where n is the order of diversity. 

r ' 

Dual and quadruple-diversity curves are similarly displaced. Note 
that maximal-ratio combiner performance is assumed. 



-67- 

In the above derivations no allowance was made for degradation of 
the signal-to-noise ratio in the Multiplex unit; the effect of this de- 
gradation is to increase the required signal-to-noise ratio at the 
receiver output. In the absence of information on this degradation 
factor, it may be assumed to be approximately 1 to 2 db for SSB-SC 
frequency division multiplex equipment. 

Procedures for using the information in figure 4-2, to estimate 
the performance of SSB systems, are outlined in Section 6 for a 
Rayleigh-fading type of received signal and for non-diversity, dual- 
diversity and quadruple -diversity. 

4.6 Performance Characteristics of FM Radio Receivers 

The performance of an FM (frequency modulated) radio receiver 
may be defined in terms of the signal-to-noise ratio per message chan- 
nel at the receiver output versus: (a) the total available RF-signal 
power, (b) the total available noise power at the receiver front-end 
input, and (c) the radio-frequency signal spectrum bandwidth; other 
system parameters may be considered as normalizing factors. 

A method for determining the above relationships for FM systems 

* 

follows; this method is based on a combined mathematical and graph- 
ical analysis of typical characteristic curves of FM radio receivers. 
These curves consist of an upper linear portion, and a lower non- 
linear portion called the "threshold" region. In the work that follows, 
the slope and the position of the upper linear portion of the receiver 
curve were calculated and the curvature of the threshold region was 
obtained from measured results on several different types of FM 
radio receivers; however, the threshold region portion of the curve 
may als"o be calculated [Stumpers, 1948] . The position of the thres- 
hold-level point was assumed to correspond to a signal-to-noise power 
ratio (P /N ) of 1 db, at the input to the receiver limiter-discrimin- 
ator. 



-68- 
Accurate estimates of the FM radio receiver performance requires 
quantitative data on the statistical characteristics of the received RF 
signal and the noise power, at the receiver input, and corresponding 
data on the signal power and the noise power at the receiver output. 

4.6.1 FM Radio Receiver Characteristic Curves 

Throughout this work, covering the development and use of the FM 
radio receiver characteristic curves, the following ideas and assump- 
tions are used as guides: 

(a) The modulating signal is sinusoidal and its frequency is equal 
to the highest frequency in the modulating baseband signal. 
Furthermore, all of the power in the modulating baseband 
signal is contained in this single sinusoidal modulating signal. 
The work is then extended to cover receiver performance for 
modulating signals at lower frequencies. Receiver perform- 
ance is shown to be improved, above the threshold region, 
directly proportional to the square of the ratio of the highest 
baseband- signal frequency to the frequency of the modulating 
signal being considered. 

(b) The radio-frequency spectrum bandwidth required for a noise- 
type modulating signal is then calculated in terms of the results 
obtained for the conditions in (a). 

(c) Receiver-input noise is unweighted, that is, its power spectrum 
is uniform or flat. 

(d) Limiting in the radio receiver is used, preceding the frequency 
discriminator. 

(e) The radio receiver IF bandwidth, B. r , is assumed to be only 

if 

wide enough to pass the radio frequency spectrum, B /, that 

rf 

is, B = B . If "bandwidth-compression 1 ' type of receivers 
are used, then B < B . The effects of bandwidth-compression 
techniques on the system design curves are considered. 



-69- 

(f) The effects of pre-emphasis and de- emphasis techniques are 

considered. * 

(g) Companders and limiters (for speech messages) are not con- 
sidered. 

4.6. 1. 1 Linear Region of FM Receiver Characteristic Curves 

For the condition where the pre-detection signal-to-noise ratio, 
P. f /N , exceeds 10 db, it can be shown (see Appendix C, (C-15)) that, 

S B P 2 P 

oc c m /AF \ r 



x 7— x — = -T- x ,,,/^ ^ w (4.19)** 



N f p V f J 2k(T A + T )f 

oc m ^m v m ' A er m 

where, 

S = Average message-signal power level per mes- 
sage-channel, at the receiver output, in watts. 

N = Receiver output average noise power level, per 
message-signal spectrum bandwidth, in watts. 

B = Message -signal spectrum bandwidth, in c/s. 

f = Frequency of the modulating sinusoidal signal, in 
c/s. This is also the highest frequency in modu- 
lating baseband signal. 

p = Average power level of an individual modulating 
signal in the baseband signal. 

P = Total average power in the composite modulating 
baseband signal. This is the power level which 
exists at a time when a specified number of 
simultaneously-active message signals are being 
transmitted. 

-^Acceptable system performance may be obtained in the "threshold 
region" where pre-emphasis and de-emphasis do not improve system 
performance, see Section 6. 

##For a discussion of the term (T + T ) and the effects of impedance 
mismatching in the receiving system, see Appendix A. 



-70- 

AF = Peak radio-frequency carrier deviation, in c/s, 
for a single sinusoidal modulating signal, having 
an average power level P equal to the total 
average power level of the composite modulating 
baseband signal. 

P = Total available carrier- signal power at the input 

terminals or "front-end" of the FM receiver, in 

watts . 

-23 
k = Boltzman's constant = 1.3804 x 10 Joules per 

degrees, Kelvin. 

T . = Receiving antenna-system output noise tempera- 
ture, degrees Kelvin; see Appendix A. 

T = Radio receiver effective input noise temperature 
[IRE Comm. on Noise, 19&0J , degrees Kelvin. 

AF _ . . 

= Deviation ratio. 



f 
m 



P and S /N are to be considered as the "variables" in (4. 19); 
r oc oc 

A F/f is the "variable-parameter" and the remaining terms in this 
m 

equation are considered as "scale-normalizing factors". 

Since (4. 19) was developed for the purpose of calculating the 
performance of FM radio receivers, a discussion of the limitations 
and application of this equation is appropriate. It should be noted that 
S /N __ is the receiver output signal-to-noise ratio for a particular 
message signal having a power spectrum centered at a frequency of f 



m 



cycles per second in the baseband. Furthermore, this equation is valid 
only for values of the predetection signal-to-noise ratio, P /N , great- 
er than approximately 10 db. In other words, there is a linear relation- 
ship between the predetection signal-to-noise ratio in the FM receiver 
IF, and the signal-to-noise ratio at the receiver output, only for values 
of the predetection signal-to-noise ratio exceeding approximately 10 db. 
In this respect (4. 19) can be deceptive because it does not include the 



-71- 
non-linear threshold region. Hence, (4. 19) should be used only to 

locate the position and the slope of the FM radio receiver character- 
istic curve above the threshold region. 

The position of the "threshold-level" point of the FM radio receiver 
characteristic performance curve is determined from the relationship 



P., P P 

if r r 



N if k(T.+T )B.- k(T A +T )f x<j/ 



where, 



A er if A er m \ f 

m 



= 10 (4.20) 



<j> (AF/f ) = Function of A F/f , calculated [ Hund, 1942] and 
plotted in figure 3-1 for "sinusoidal- signal" FM 
modulation. This function gives the number of 
significant sidebands in the radio frequency- 
carrier spectrum; assuming that the modulating 
signal is a single sinusoidal voltage wave at the 

highest frequency f , in the modulating baseband 

, m 

signal. 

Equation 4.20 was used to determine the "threshold level" points 

on the curves in figure 4-3; these points are the limits at which the 

curves cease to be linear for decreasing value of P . The slope of the 

r 

upper linear portion of the receiver characteristic curves, shown in 
figure 4-3, was determined from (4.19 )• 

4.6.1.2 Threshold Region of FM Receiver Characteristic Curves 

The shape or curvature of the non-linear "threshold region" of 
the radio receiver characteristic curve was obtained from averaged 
measured data on several different types of conventional FM radio 
receivers. Calculated threshold-level points are shown in figure 4-3 
as the intersections of the threshold-level curve and the receiver- 
characteristic curves. 



-72- 

FM RADIO RECEIVER CHARACTERISTIC CURVES 
STEADY RECEIVED SIGNAL 




: 230 -225 -220 -215 -210 -205 -200 -195 -190 H85 -180 -175 -170 -165 H60 H55 
10 Log |0 P r - 10 Log |0 f m -IO Log (T A +Ter) 

ivV'e 

Figure 4-3 



S_ AVERAGE OF THE TOTAL SIGNAL POWER.WATTS AT RFnriwre , NPLrr 

fm(T A + T e r) "AVERAGE OF THE TOTAL RELATIVE NOISE POWER' Al ™ lulllv,in MNru ' 






-73- 
From the above outline it is possible to determine the shape and 
the position of an FM radio-receiver characteristic curve, in terms of 

the receiver -input RF-signal power to the receiver-input relative noise 

P 
r 
power ratio, r— p= — — — — ; and the receiver output (message) 

A er m 

channel- signal power to channel-noise power ratio, S /N ; for a 

oc oc 

given set of system parameters. 

4.6.2 FM Radio Receiver Input-Power and Bandwidth Require- 
ments for Steady Received Signals. 

Following is a description of the method used to construct the FM 
radio receiver curves shown in figure 4-3, using the scale normalizing 
parameters as noted in the abscissa and in the ordinate titles. The 
shape of these curves and their locations, relative to the absicssa and the 
ordinate scales and relative to each other, are important if they are 
to be used to obtain accurate determinations of the required bandwidth 
B A = B. f ) and the receiver input-power, P , required for particular 
grades of service, or for particular values of receiver -output signal- 
to-noise ratios, S /N 

oc oc 

The non-linear threshold-region portions of the curves were ob- 
tained from averaged measurements on a number of FM radio receivers 
of different types, as indicated by the measured points shown in figure 
4-3. It was assumed that the threshold- region curvature of the curves 
did not vary appreciably with change in the deviation ratio AF/f , 
hence all curves shown in figure 4-3 are identical in shape. 

The positions and the slopes of the upper (linear) portions of the 

FM radio receiver curves in figure 4-3 were calculated, using (4.19) 

and (4. 20); each of the scale-normalizing factors, B , f , and 
x ' ° c m 

P /p were set equal to unity. T was set equal to the standard 



-74- 

temperature, T (= 290 degrees Kelvin); this condition is equivalent to 
o 

the assumption that the modulated RF signal was obtained from a signal 

generator with an output impedance temperature of 290 degrees Kelvin. 

T was assumed to be degrees Kelvin; this assumption corresponds 
er 

to a receiver noise figure of unity, or db. Note that A F/f is con- 
sidered here as the "variable" parameter and not as a scale-normal- 
izing factor. The upper linear portions of the curves are displaced 

relative to each other on the ordinate scale an amount which is depend- 

/ 2 
ent upon the relative values of ( A F/f ) . 

m 

The threshold-region section of the receiver curves are drawn 

tangent to the upper linear portions of the curves, at the points given 

by (4. 20); these points correspond to an IF signal-to-noise ratio 

p /N =10. This value of 1 db for P. r /N. r , applies only to the case 
if if if if 

where the received radio-frequency signal is non-varying or steady. 

The curves in figure 4-3 were drawn for a value of f assumed to 
6 m 

be equal to the highest frequency in the modulating composite baseband 
signal; therefore, the receiver output signal-to-noise ratio values, for 
the linear portion of these curves, would be increased by a factor, 

(f /f ) , or 20 Log — — db, for modulating signals of frequency f - 

c 
-where f < f . However, when minimum receiver-input power P is 
cm r 

the criterion, the system parameters should be adjusted so that the 

receiver will be operated in its threshold region for a small percentage 

of the time; under these conditions, when P is above the threshold- 

r 

level value, the receiver output signal-to-noise ratio exceeds the 

required S /N value and pre-emphasis techniques are of doubtful 
oc oc ^ 

value. In the threshold region, the receiver output signal-to-noise 

ratio, S /N , approaches the same value for all frequencies in the 
oc oc ^ 

modulating baseband signal; see figure 4-8 and Section 4.6.4. Hence, 
the curves in figure 4-3 will yield accurate results in the threshold 
region and conservative results above the threshold level. 



-75- 

The radio-frequency spectrum bandwidth, B , associated with 

rf 

particular values of AF/f and f (see (C. 8)) is given by, 



B = f x ^AF/f ) (4.2 1) 



rf m 



m 



where, 

B = Radio-frequency spectrum bandwidth which 
includes all components or sidebands having 
amplitudes equal to or greater than one percent 
of the unmodulated carrier amplitude. 

The relationship between AF/f and 4> (AF/f ) was calculated 

m m 

[Hund, 1942] and is shown in figure 3-1, for sinusoidal- signal modu- 
lation. Values for B /f , associated with particular values of AF/f , 

ri m m 

were obtained by combining (4.2 1) and figure 3-1; these associated 

values of AF/f , B Ji and d> (AF/f ) are shown in figure 4-3. 
' m rf m v ' m 6 

The receiver characteristic curves in figure 4-3 were re-plotted 
in a more convenient and useful form, as shown in figure 4-4. 

The interrelationships between three important factors in an FM 

radio communication system are shown in figures 4-3 and 4-4. These 

factors are: ( 1 ) required bandwidth or radio frequency spectrum, B ., 

ri 

(2) grade of service or minimum permissible receiver output signal- 
to-noise ratio, S /N , and (3) required radio frequency carrier 
oc oc 

signal power, P , relative to the noise-power level, k (T . + T )f , 
& * r ^ A er m 

at the receiver input. 

The designer of a radio communication system is usually given 

the specified number of voice channels, that is, the baseband spectrum 

bandwidth, from which he can determine f ; the grade of service or the 

m 

maximum tolerable average message error rate is also specified, from 

which the system designer can establish the required receiver output 

signal-to-noise ratio, S /N . The initial problem, is to choose 

oc oc 

the required radio-frequency carrier spectrum bandwidth B (where 



-76- 



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-77- 
FM systems are involved), and the required radio frequency carrier- 
signal power level, P , at the receiver input. The curves in figures 
4-3 and 4-4 make it possible to select required sets of values for B 

and P , for a particular grade of service, or S /N , for specified 
r oc oc 

types of messages. The curves in figure 4-4 can also be used as guides 

when it is desired to trade or interchange required P , for required 

r 

bandwidth, B r . 
rf 

The required deviation ratio, A F/f , associated with the required 

m 

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be obtained from the right-hand scale in figure 4-4. Hence, the requir- 
ed deviation AF can be obtained for a given baseband bandwidth B, or 

& b 

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m 

figures 4-3 and 4-4 applies only to the case of a steady or non- varying 
radio frequency received signal. Procedures for using the design 
curves in figure 4-4 are outlined in Section 6. 



4. 6. 3 FM Radio Receiver Input-Power and Bandwidth Require- 
ments for a Rayleigh-Fading Type of Received Signal. 

FM receiver characteristic curves, were obtained for a Rayleigh- 
fading type of received signal and for non-diversity, dual diversity, 
and quadruple diversity by graphically combining the curves in figure 
4-3 (for steady received signal) with the appropriate maximal-ratio 
Combiner -output distributions [ Brennan, 1958] shown in figure D-lb; 
details of the method used to combine the FM receiver performance 
characteristic for steady received signal with the Combiner character- 
istics are given in Appendix D. The three sets of curves obtained by 
this procedure were then scaled and replotted as shown in figures 4-5, 
4-6 and 4-7; for non-diversity, dual diversity, and quadruple diversity, 
respectively. 



78- 



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The major differences between the design curves in figures 4-4, 

4-5, 4-6 and 4-7 are that the P values for the steady received signals 
are average values while the P values for the fading type of received 
signals are median values of the non-diversity average-power levels of 
the received signal. The positions of the curves, relative to the ab- 
scissa and the ordinate scales, are of course different for the different 
orders of diversity; furthermore, their relative positions are also 
dependent upon the type of combining and the actual performance of the 
Combiner. This latter point greatly influences the accuracy of system- 
performance estimates based on the design curves which are developed 
in this report, compared to actual system-performance measurement 
results. Procedures for using the design curves in figures 4-4, 4-5, 
4-6, and 4-7 are outlined in Section 6. 

4.6.4 Pre-emphasis-De-emphasis Techniques 

In FM systems, the rms noise voltage in the receiver -output base- 
band signal is proportional to the frequency in the baseband- signal 
spectrum. If equal power levels are assigned to each spectral compo- 
nent in the modulating baseband signal at the modulator input, regard- 
less of the positions of the modulating signals in the baseband signal, 
it will be found that the receiver-output signal-to-noise ratio will vary 
as shown in figure 4-8. 

Pre-emphasis of the spectral components in the modulating base- 
band signal, proportional to the frequency of the component in the base- 
band, and a corresponding de-emphasis of the baseband components in 
the radio receiver, result in an equilization of the radio receiver out- 
put signal-to-noise ratio for all components in the baseband signal. 
However, the above conditions apply only when the radio receiver is 
operating above its "threshold level"; below the receiver threshold 



-82- 

FM RECEIVER CHARACTERISTIC CURVES, WITHOUT PRE-EMPHASIS 



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Fig. 4-8 



-83- 
level the use of pre-emphasis-de-emphasis techniques results in a deg- 
radation of receiver-output signal-to-noise ratio. This condition is 
evident from figure 4-9. Hence, pre-emphasis-de-emphasis techniques 
will only be effective when the FM receiver is operated above its thres- 
hold level. 

FM system performance may be estimated without pre-emphasis- 
de-emphasis techniques as outlined in this paper; the results will be 
accurate in the receiver threshold region and will be conservative 
above the threshold level. Final design estimates may be modified, in 
terms of the baseband- signal frequency, by the use of data similar to 
that shown in figure 4-8; however, there may be some question as to 
the practicability of this procedure. 

4. 6. 5 Bandwidth Compression in FM Receivers. 

All of the evidence to date shows that the various proposed receiv- 
er bandwidth-compression schemes have the effect of lowering the 
"threshold-level" of the FM receiver, relative to the received carrier- 
signal power and the total noise power "accepted" by the receiver. The 
accepted noise power is that portion of the total receiver-input noise 
power which appears at the input to the FM receiver limiter-discrimi- 
nator, or demodulator unit. It should be noted that the position of the 
threshold level also depends upon the characteristics of the limiter and 
hence the threshold level may, in some cases, possibly be lowered by 
improvements in the limiter characteristics. 

The overall effect attained by compressing the receiver bandwidth 
is shown in figure 4-10. It is evident that there is no enhancement of 
the bandwidth-compressed receiver output signal-to-noise ratio above 
the threshold level point; this situation is due to the fact that the pre- 
detection frequency deviation, AF, is reduced when the bandwidth is 
compressed, and hence the net effect is that there is no improvement 
in the receiver performance above the threshold level. 



-8i+- 



EFFECT OF PRE-EMPHASIS ON FM RECEIVER CHARACTERISTIC CURVES 



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Fig. 4-9 



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EFFECT OF BANDWIDTH COMPRESSION ON 
FM RECEIVER CHARACTERISTICS 



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COMPRESSED BANDWIDTH 



RELATIVE RF SIGNAL- POWER LEVEL AT RECEIVER INPUT, dbw 

Fig. 4-10 



-86- 
The FM- receiver design-curve data presented in this report may 
be modified to include receiver bandwidth- compression or modified- 
limiter effects by incorporating the data from figure 4-10. The data 
shown in figure 4-10 may be obtained by either (1 ) performing the 
indicated comparative measurements on the FM receivers, for the 
conditions shown in the figure, and for identical frequency-modulated 
RF signals at the input terminals of the radio receivers; or (Z) by 
estimating the value of the bandwidth compression factor, K , and 
modifying the design curves as shown in figure 4-10 or figure 4-11. 

The factor K is defined in terms of the compressed-bandwidth 
c 

receiver threshold-point power level, P 1 relative to the similar 

power level, P", for an un-compressed-bandwidth receiver. The 

value for K may be obtained from the measured characteristics of 
c 

the receivers, as shown in figure 4-10. 

The modified design curves, well below threshold and for high 

values of B r /f , will parallel the original curves and will be shifted 

to lower values on the relative P scale; this decrease in scale values 

r 

will be equal to K , in db. The modified curves will converge with 
and will join the original design curves at the original threshold-level 
points. Above the threshold-level points, the modified design curves 
are identical to the original design curves- -which apply to FM receivers 
without bandwidth compression. The effect of receiver-bandwidth com- 
pression on the shape of the FM system-design curves, is shown in 
figure 4-1 1. 



-87- 



EFFECT OF BANDWIDTH COMPRESSION ON 
FM SYSTEM DESIGN CURVES 



CQ 



K c ,db 




THRESHOLD LEVEL 



RELATIVE RF SIGNAL-POWER LEVEL AT RECEIVER INPUT, dbw 

Fig. 4-11 



-88- 
5. PERFORMANCE CHARACTERISTICS OF MESSAGE-SIGNAL 

DECODER UNIT 

The performance of the message-signal decoder unit is defined 
here in terms of the relationship between the message-channel signal- 
to-noise ratio, S /N . at the decoder-unit input, and the error rate 
oc oc 

in the message at the decoder-unit output. This definition of the mes- 
sage-signal decoder unit performance is general and may be applied to 
any type of decoder, provided that proper consideration is given to the 
statistical characteristics of both the message-channel signal and the 
message-channel noise at the decoder-unit input, and the resultant 
message error rate. 

At the decoder-unit input, both the short-term message-channel 
signal power, S , and the short-term message-channel noise power, 

N , will vary independently with time; furthermore, the distributions 
oc 

of S and N are dissimilar. Hence, in order to obtain measurable 
oc oc 

and definable results it is necessary to analyze the performance of the 

message- signal decoder unit in terms of the average signal power-to 

average noise power ratio, S /N , at the decoder input. The 
2 — oc oc 

problem is further complicated by the fact that the ratio S /N varies 

oc oc 

with time and hence its distribution must be considered; finally, the 

distribution of the noise power affects the message error rate, for 

particular values of S /N , as shown in figures 5- 1 and 5-2. 

oc oc 

In this paper the decoder-unit performance is determined, for 
the above conditions at the decoder input, as follows: 

(1) The decoder-unit performance is either estimated or measur- 
ed for a range of values of the average signal power-to-average 

noise power ratio, S /N , at the decoder input. S /N 

oc oc oc oc 

is assumed to be non-time varying or steady, and the decod- 
er-input noise, N , is assumed to be "white". The 
oc 



-89- 



CUMULATIVE AMPLITUDE DISTRIBUTION OF NOISE AT RECEIVER OUTPUT 

FOR RECEIVED CARRIER POWER LEVELS ABOVE AND BELOW THE 

FM RADIO RECEIVER THRESHOLD LEVEL 



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PERCENTAGE OF TIME ORDINATE VALUES ARE EXCEEDED 



Figure 5- I 



-90- 



MEASURED RELATIONSHIP BETWEEN THE STEADY -SIGNAL POWER TO 

MEAN NOISE POWER RATIO, AT THE TELETYPE RECEIVER INPUT TERMINALS, 

AND THE AVERAGE TELETYPE CHARACTER ERROR RATE 



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MEASURED AT TELETYPE RECEIVER (DECODER -UNIT) INPUT 



db 



Fig. 5-2 



-91- 
"measured steady signal" curve shown in figure 5-3 is typical 

for narrow-band FSK teletype decoder-unit performance - 

(2) The performance of the decoder unit, for time-varying 
(decoder-input) signal-to-noise ratios, is then estimated by 
combining the above steady- signal decoder-unit character- 
istic with the distribution of the time-varying decoder-unit 
input signal-to-noise ratio. Details of this procedure are 
given in Appendix D. 2; results are shown in figure 5-3 for 
non, dual, and quadruple diversity, assuming maximal- ratio 
combining. 

(3) Under conditions where "impulse -type" of noise appears at 
the decoder-unit input, the procedures outlined in (1) and (2) 
above would he followed; the resultant decoder-performance 
curves would be similar to those shown in figure 5-3 (for 
white noise at the decoder input) but they would be shifted to 
higher values of S /N , for given values of message error 
rates. See figures 5-1 and 5-2. 

The shapes and the positions of the curves in figure 5-3 are 

determined by the distribution of the signal-to-noise ratio, S /N , 
y B oc' oc 

at the decoder-unit input; in turn, this distribution (for our purpose) 
is dependent upon: (a) the distribution of the received RF signal at 
the radio receiver input, (b) the radio-frequency receiver character- 
istics, (c) the order of diversity, and (d) the combiner characteristics. 
Combiner performance was assumed to be as shown in figure D-lb, for 
a Rayleigh-fading type of radio-frequency carrier signal, using maxi- 
mal-ratio combining, and for SSBSC and FM radio receiver character- 
istics as given in Sections 4.5 and 4.6. 

The principal requirement, which may be used as a guide when 
designing and testing radio communication systems, is that the (receiv- 
ed) average message error rate should not exceed a specified value for 



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-93- 

particular message-load conditions on the system. Message errors 

usually occur in bursts or groups, and are associated either with 
signal fades or noise peaks, or both. Hence, the "short-term message 
error rate" will vary from zero to some comparatively high value 
during the sampling period; therefore, the term "average message 
error rate" is used throughout this paper, in order to differentiate 
between the short-term message error rate and the message error 
rate which is obtained by averaging the short-term message error 
rates over the entire sampling period. For example, the average 
teletype character error rate is defined here as: 

Total number of teletype character errors occurring during 

sampling period 
Total number of teletype characters transmitted during the 

sampling period 

The interval of time included in the sampling period is subject 
to the mutual agreement of all parties involved in the design and 
acceptance-tests of the radio communication system. 

Narrow band FSK (frequency- shift-keyed) teletype equipment 
employing a frequency shift of approximately ± 40 cps is generally 
available in multichannel tropospheric radio communications systems, 
and considerable data are available on the correlation between the 
teletype character error rate and the signal-to-noise ratio at the 
teletype-receiver input or in the received baseband signal. Hence, 
teletype character error rates may be used conveniently as the cri- 
teron in designing and testing radio communication systems. Other 
types of messages, such as voice, digital data, or facsimile, could 
be used of course, provided that the relationship between the message 
error rate, i.e., grade of service, and the signal-to-noise ratio at the 
dec oder -unit input is known. 



-94- 
For a specified allowable teletype -message error rate, from the 
data shown in figure 5-3, we can determine the required message- 
channel signal-to-noise ratio, S /N , at the radio receiver output; 
° oc oc 

proper corrections are required for the effect of the message-signal 
multiplexing equipment on this signal-to-noise ratio. Similar informa- 
tion is required for the other types of messages being transmitted. 
These required radio-receiver output signal-to-noise ratios are then 
used in (4.9), Section 4, and with the proper set of system-design 
curves in figures 4-2, 4-4, 4-5, 4-6 or 4-7, to determine correct 
values for P , B , AF, etc. The method of applying the above pro- 
cedures is outlined in Sections 4.5 and 4.6. A typical design problem 
for SSBSC-AM and FM radio communications systems is solved in 
detail in Section 6. 

From figures 5-1 and 5-2, it is evident that the performance of 
the FSK Teletype Decoder is not influenced, to a large degree, by the 
statistical characteristics of the noise at the decoder input terminals. 
This situation is due primarily to the effectiveness of the decoder 
limiter in removing the "amplitude" noise, which is associated with 
the signal. FM decoders for other types of messages would be expec- 
ted to give results similar to the results obtained above for FSK tele- 
type messages. 

There is a need for up-dating voice-message performance data, 
similar to that given in figure 5-3 for teletype message errors vs. 
signal-to-noise ratios. In the system-design example worked out in 
section 6, the required voice message signal-to-noise ratio was ob- 
tained from the work of French and Steinberg, 1947; Eagan, 1944; 
and Beranek, 1947. An alternative solution to this problem is to 
accept the required voice message signal-to-noise ratio specified by 
the user of the system. 



-95- 
6. RADIO COMMUNICATION SYSTEM DESIGN AND TEST 

PROCEDURES 



A typical system-performance problem is solved in this Section, 
in order to indicate the procedures to be followed when using the system 
design curves to estimate the following major factors: (A) average- 
power level of the total modulating signal, P , (B) required peak 
frequency deviation, AF (for FM systems), (C) required radio- 
frequency spectrum bandwidth, B , and (D) required receiver-input 

ri 

RF-signal power level, P . The "required" values of these factors 
are associated with a specified maximum message error rate or 
grade of service. For the condition where certain parameters and 
factors such as B ., transmitter power, P , antenna gain, G , mes- 
sage load, etc. are given, the design curves may also be used to 
estimate the grade of service which will be provided by the system. 
Calculations are carried out for SSBSC-AM and for FM systems using 
dual diversity and maximal-ratio combining, and assuming a Rayleigh- 
fading type of received RF signal; for other orders of diversity the 
analysis is identical using the appropriate design curves. 

It is assumed that the following system specifications are given: 

(1) Number of available message (voice) channels. 

(2) Specified number of simultaneously-active message channels. 

(3) Number of messages of each type, M , and bandwidth, B , 

J. *— 

required for each message signal. 

(4) Maximum allowable average message error rate. 

(5) Type of Multiplex equipment to be used, and the baseband 

spectrum bandwidth required to accomodate the specified 
number of available message channels. 

(6) Transmission losses in the transmitting and receiving 
antenna systems. 



-96- 

(7) Antenna noise-power temperature, T . 

(8) Receiver noise figure, F , or receiver -input temperature, T 
The procedures for using the design curves in this report are as 

follows: 

(a) Determine the required minimum value of the receiver output 
average signal power-to-average noise power ratio, S /N , 
associated with the specified maximum allowable average 
message-error rate, for a specified order of diversity. For 
teletype messages this information may be obtained from 
figure 5-3. For other types of messages, data similar to 
that given by figure 5-3 would be required, and could be 
developed using methods outlined in Appendix D. 

(b) Determine P /p , the message-signal power-loading ratio 
in the modulating baseband signal, using (4. 9) and associated 
information. Where p is the individual message-signal 
average modulating power and P is the total average mes- 
sage-signal modulating power in the modulating baseband sig- 
nal, measured at a time when the specified number of simul- 
taneously-active channels are operating. The maximum 
number of simultaneously-active channels should be speci- 
fied, otherwise the designer is compelled to choose a value 
for this important factor. 

(c) Determine the highest modulating frequency, f , from the 

specified number of available voice channels, or from the 

modulating baseband- signal bandwidth. The value of f will 
66 m 

depend upon the type of multiplex system being used; f is 
always equal to the highest frequency in the modulating base- 
band signal. 



-97- 

(d) Calculate the antenna- system temperature, T , and the ef- 

fective receiver-input temperature, T , from specified 

er 

receiving-system parameters: LLL ,T,T,T,T , 

c t du act du 

and F . See Appedix A. 
r 

The above information on the system parameters is combined with 
the data in the appropriate equipment-performance design curves, in 
order to determine the required RF bandwidth, B , and the required 
receiver-input RF signal power level, P ; for the specified grade of 
service or permissible message error rate. 

6.1 Determination of the Basic System Parameters: 

S /N , B , P /p , f , T . , and T 
oc oc c m m m A er 

for Specified Average Message Error Rates, 

Number of Available Message Channels, and 

Specified Receiving-System Parameters. 

This Subsection outlines the method of calculating the information 

required in items (a), (b), (c), and (d) in Section 6, from the system 

specifications which are outlined in Items (1 ) through (5) in Section 6. 

The parameters S /N , B , P /p , and f obtained from these 

oc oc c m m m 

calculations apply to both SSBSC-AM systems and FM systems. A 
specific example is considered in order to indicate more clearly the 
design procedures, and the calculated values of the above parameters 
are then used in Subsections 6. 2 and 6.3. 

Radio Communication System Specifications 

Number of available voice channels 36 

Maximum (specified) number of simultaneously- active 

channels 9 

M = Maximum number of teletype message channels 
simultaneously-active (16 teletype messages per 
voice channel) 32 



-98- 



M = Maximum number of simultaneously -active 
voice-message channels (continuous talkers) 

Bandwidth B required for each teletype signal 

Bandwidth required for each voice signal 



7 
110 c/s 
3200 c/s 



Maximum allowable average teletype character 
error rate 

Type of Multiplex 

Multiplex baseband bandwidth, 4 x 36 = 

Type and order of Diversity 



Transmission-line and receiving-antenna 

loss factor, L L L , 
c t du 

Antenna noise temperature, T 



Receiver noise figure, F 



0. 01% 
SSBSC, Frequency div. 

144 kc 
Space, Dual 

1.26(=ldb) 
300 deg. K 
5 db 



Calculations 

From figure 5-3, the required signal-to-noise ratio at the tele- 
type-message decoder unit input, for dual diversity operation, is 28 db; 
and allowing 2 db degradation of the signal-to-noise ratio through the 
Multiplex unit, we have 



lOLogfS /N / telet VP e ^ 
10 oc oc ^message J 



= 3 db 



and, 



S /N f telet YP e V 10 00. 
oc oc y messagey 



From the system specifications listed above we have, 
B (teletype) = 110. c/s 



B (voice) = 3200 c/s 
c 



-99- 
The message-signal loading ratio in the modulating baseband sig- 
nal, P /p ., is determined from (4. 9), see p. 56 . This equation 



m mi 



.th 



shows that P /p . for the i — message signal depends upon the requir- 
ed receiver-output signal-to-noise ratios, S /N , for each of the 

oc oc 

message signals in the modulating baseband signal. The required 
decoder-input signal-to-noise ratio for teletype message signals was 
obtained from figure 5-3, and was found to be 28 db. The required 
decoder-input signal-to-noise ratio for voice message signals (dual 
diversity) is assumed to be 33 db [French and Steinberg, 1947; Eagan, 
1944; Beranek, 1947] . The estimated message intelligibility for 
voice messages, corresponding to a signal-to-noise ratio of 33 db, is 
90 percent. Therefore, in the receiver baseband-signal at the receiver 
output, we have, 

S 



oc / voice 



N 



oc 



message 



= 33 + 2 = 35 db = 3160, 



/ 



Substituting the above information in (4.9) p. 56, we obtain, 



P 
m /per teletype \ _ (32 x 110 x 1000) + (7 x 3200 x 3160) 

p . I message )' 110x1000 

mi K ° / 



and, 



10 Log 



10 



m 



m 



per teletype 
message 



) = 10 Log 1() (677.) = 28.3 



db 



In systems which require the transmission of pilot tones, it is 
necessary to include the pilot-tone modulating signal power levels, p , 

m 

and the number of pilot tone signals, M, in the above calculations. The 
net effect of the inclusion of these pilot tone- signals will be to require 
an increase in the transmitter -output power; in order to transmit the 
pilot-tone signals without altering the grade of service of the "useful" 
message load. 



-100- 

The value of f is equal to the highest frequency in the modulating 
m 

baseband signal and is, 



f = 12 + (36 x 4) = 156. kc 
m 



or, 



10 Log, rt f = 52. db, 
s 10 m 



Note that the lowest-frequency in the multiplex is 12 kc and that 
each voice channel occupies a bandwidth of 4 kc in the modulating 
baseband signal including the guard bands; this simple relationship 
applies only to SSB frequency division type of Multiplex systems. 

The value for T is obtained from either (4. 3) or (4. 3a), p. 37. 

Using (4.3a), and assuming P /k B L to be 200 degrees Kelvin 

and T = 300 degrees Kelvin 
a 6 

T - T 

T a = ~r — % T- 2 - + T + P, /kB. r L 

A L L L. o it' if D 

c t du 



300 - 290 

— 6 7 + 290 + 200 = 498 deg K 



The receiver noise figure F is assumed to be 5 db or, F = 3. 16; 

r r 

therefore, from (4.6), p. 38. 



T = (3. 16 -1) 290 = 625 degrees Kelvin 



er 



*Note that the required value of the Duplexer isolation factor , L , 

depends upon P. , k, B.. and the allowable level of the Local-Trans- 
it ii 
mitter "noise temperature", at the Duplexer output. 



-101- 

6.2 Equipment Design Procedures, SSBSC-AM Systems, 

For a Rayleigh-Fading Type of Received Signal, 
Dual Diversity, and Maximal-Ratio Combining 

It is required to determine P from the SSBSC-AM radio-receiver 

r 

performance curves in figure 4-2; using the above listed specification 
items (6), (7), and (8), and the values of the parameters which were 
derived in Subsection 6. 1. 

For SSBSC-AM systems, the modulator-demodulator conversion 
factor, D , may be obtained from figure 3-5. Assuming D = 4 db, 

o o S S 

the (noise-type) modulating- signal power level, P , (= P ) required 

mn m 

so as to insure that 100% amplitude -modulation conditions will be 

exceeded for only 1. 2 percent of the time, will be 4 db below the power 

level, P (100%). The value for P (100%), corresponding to 100% 
ms ms 

amplitude modulation, may be obtained from the modulator character- 
istic, which is in terms of sinusoidal moi 
versus percent amplitude modulation, m 



istic, which is in terms of sinusoidal modulating- signal power, P , 



a 
The power level, p . , of the individual message-signal, at the 



mi 



modulator input, may now be obtained from the previously calculated 

value for P /p . (= P /p .). Note that this calculated value for 
m mi mn mi 

p . applies only to the case where the maximum number of simultan- 
mi 

eously-active message channels is one-fourth of the available channels, 
as specified in this example. Other specified message-loading con- 
ditions will require different values for p 

^ mi 

From the above derivations we have 



10 Log lrt D = 4 db 
10 ss 



10Log in P =10Lo gl „P (100%)- 4 db 
& 10 mn 6 10 ms 

10Log 1A p . =10Log in P -28.3db 
& 10 *mi 6 10 mn 



-102- 

Note that both P and p . are dependent upon the measurable or 
mn mi 

estimated value of P (100%), obtainable from the modulator -per- 
ms 

formance characteristics. 

The next step in the design procedure is to determine the operating 

value, on the normalized ordinate scale in figure 4-2. This scale value 

is obtained by substituting the above information on S /N , B , 

P /p ., and f in the "title" of the ordinate scale, which is, 
m mi m 

10Log 1/x ( t^- J + 10Log lrt B - lOLog, f +10Log lrt (^^ 

S 10VN J 5 10 c 5 10 m 10Vp • 

oc mr 

or, 

lOLog^dOOO)^ 1 ^) ♦ 10Log 10 (110) - 10Log 10 (156000) 

+ lOLog (677. ) = 30 + 20.4 - 52. + 28. 3 = 26. 7 db 

From figure 4-2, using the above calculated ordinate -scale value 
of 26 . 7 db and the dual -diversity curve, we obtain a value of -207- 5 on 
the normalized abscissa-scale. Therefore, 

10 Log 10 P r - 10 Log 1() £ m - 10 Log 10 (T A ♦ !„! = -207. 5 

Substituting the values of T . and T (and f =52 db) in the above 
& A er m 

expression for the abscissa- scale value, we obtain for the SSBSC-AM 

system, 

10 Log.. P =52+10 Log,„ (498 + 625) - 207. 5 = -125.1 dbw 
10 r °10 

However, this value of P (-125.1 dbw) must be corrected for the 

r 

effects of system intermodulation noise; see section 4. 3. 6. Assuming 
that the receiver -output system intermodulation noise is equal to the 
receiver -output noise due to the pre-detection amplitude noise, the 

required correction to the above calculated value of P is 3 db. Hence, 

r 



-103- 

10 Log. A P = -125. 1 + 3 = -122. 1 dbw (SSBSC-AM) 
1 U r 



The value of P (-122. 1 dbw) is the median non-diversity received 

r J 

(total) RF signal-power level at the SSB receiver input, required to 

transmit the specified message load, with an average teletype -message 

error rate of . 01 percent and a predicted voice-message intelligibility 

of 90 percent. This value of P is also the value to be substituted in 

r 

(7. 1) of Section 7, together with appropriate values of the other factors 
in (7. 1), to obtain the required transmitter -output average power level, 

p f 

In a SSBSC-AM system the radio -frequency spectrum bandwidth is 
approximately equal to the highest frequency in the modulating base- 
band signal; therefore, 

B., = B , = f =156 kc. 
if rf m 

6.3 Equipment Design Procedures, FM Systems, for a 

Rayleigh- Fading Type of Received Signal, Dual 

Diversity, and Maximal -Ratio Combining 

It is required to determine B and P from the performance 

rf r 

curves for FM systems using the system specifications and the values 
of the derived parameters in Subsection 6. 1. Note that for the case 
of an FM receiver which does not use bandwidth compression, the 
"bandwidth compression factor" K (figure 4-10) is equal to 1 and, 

B rf = B if • 

The first step in the design procedure is to ascertain which one of 

the design-performance curves in figure 4-6 p. 79 (dual diversity), is to 

be used. The particular design-performance curve to be used is found 

by substituting the above calculated values for S /N , B , f , and 
' ° oc oc c m 

P /p . in: 
m mi 

10 Log (S /N )+ lOLog, B - 10 Log f + lOLog, (P /p ) 
& 10 oc oc 10 c s 10 m 6 10 m *m 



-104- 

This procedure is identical to that used in the case of SSBSC-AM 

design calculations, except that the factor D (= P (100%)/P ) is 
& c ss ms mn 

not involved for the FM case. A corresponding factor, P /P , is 

ms mn 

derived at the point in the FM- system design where the bandwidth ratio, 

B Ji , is chosen. Therefore, making the proper substitutions in the 
rf m 

above expression we obtain, 
10Log 10 (1000^f g ^) + 10Log 10 (110)-10Log 10 (152000) + 10Log 10 (677., 

= 30 + 20. 4 - 52 + 28. 3 = 26. 7 db 

The particular design-performance curve in figure 4-6, which has 

a "scale-value" of 26. 7 db, is now used to determine B . and P . Any 

rf r - 

point on this curve yields a combination of values of B Ji and 
ri m 

P /f (T . + T ) such that the average teletype character error rate 
r' m A er 6 yt ^ 

is . 01 percent, for the above values of f and P /p . The system 

m m m 

designer may now decide to what extent he should interchange or trade 

bandwidth, B - or B.., for required receiver-input power, P , that is, 
ri if r 

he may choose any point on the "26. 7 db" design curve in figure 4-6 
as the operating point. 

For operation at the receiver threshold-level point on the "26. 7 db" 
design curve in figure 4-6, we have, 



B Ji =10.5 
rf m 



or 



B . = 10.5 x 156 = 1638 kc 
rf 



-105- 

and 

A F/f =2.8 
m 



or 



AF = AF = 2.8 x 156 = 436.8 kc 

s 



The required total modulating- signal average power level, P , 

ms 

may be determined from the above required value of A F (= 436.8 kc) 

s 

and the measured or the estimated modulator characteristics, AF 

s 

versus P , where the modulating- signal is a single sinusoidal signal. 

The required modulating- signal power level, P , for a white-noise 

mn 

modulating signal, may be obtained from figure 3-4, relative to P , 



using B Ji - 10.5; that is, 
rf m 



ms 



10 Log P =10Log. P -5.5db 
10 mn 10 ms 



The correct power level, p ., of the individual message-signal, 
at the modulator input may now be obtained from the previously- calcu- 
lated value of P /p . (= 28. 3 db) or, 
m mi 



10 Log, ^p . = 10Log in P -28.3db 
5 10 Vi s 10 mn 



Note that this calculated value for p . applies to this specific example, 

mi 

where it was specified that the maximum number of simultaneously- 
active channels is one-fourth of the available channels. Other message' 

loading ratios will require different values for p .. Also, P and 

mi mn 

p . are in terms of the measurable sinusoidal modulating- signal 
mi 

power level, P 

ms 



-106- 

The above calculated value of the radio-frequency signal bandwidth, 

B r , is the required bandwidth which includes 99. 99 percent of the 
rf 

spectral energy in the FM radio-frequency spectrum. B , is the same 

for either a sinusoidal modulating signal or a noise -modulating signal, 

provided that the proper values are used for P and P 

ms mn 

The selected value for B Ji (teletype-voice) = 10.5, may now 

rf m 

be used in figure 4-6 to determine the required median of the non- 
diversity receiver-input RF signal-power level, P , as follows: at 

the intersection of the B Ji - 10.5 level on the ordinate scale, and 

rt m 

the "design curve" for 26. 7 db, we have, 



10 Log 10 P r - 10 Log 1() f m - 10 Log 1() (T A+ T ] = -210. 5 



or, for the FM system 



10 Log P = 52. + 30.4 - 210. 5 = -128. 1 dbw. 



This value of P (-128. 1 dbw) must be corrected for the effects 
r 

of system intermodulation noise; see section 4. 3. 6 and figure 4-A. 

Assuming that the receiver-output system intermodulation noise is equal 

to the receiver-output noise due to the pre-detection amplitude noise, 

the required correction to the above calculated value of P is 3 db. 

r 

Hence, 

10 Log lrt P =-128.1+3. =-125. 1 dbw (FM) 
1 r 

The above calculated value of P (-125. 1 dbw) is the median non- 

r 

diversity received (total) RF signal-power level at the FM receiver 
input, required to transmit the specified message load, with an average 
teletype -message error rate of . 01 percent and a predicted voice- 
message intelligibility of 90 percent. This value of P is also the 



-107- 

value to be substituted in (7.1) of section 7, together with appropriate 

values of the other factors in (7. 1), to obtain the required transmitter- 
output average power level, P . 

Note- -The assumed value of 3 db for FM system intermodulation 
noise may be too low in a system which employs an RF spectrum band- 
width of several Mc/s. A more realistic value might be 3 db for equip- 
ment intermodulation noise and 2 db (B = 1.64 Mc/s) for path-modu- 

rf 

lation noise. With the additional correction of 2 db for path modulation 
effects, the required value for median P (FM) = - 123. 1 dbw. 

The above design procedures apply to the condition where particu- 
lar system specifications and a grade of service are given and it is 
required to determine the total average modulating- signal power level, 

P , the individual message-channel average modulating-signal power 
mn 

level, p ., the RF signal spectrum bandwidth, B r , or B. r , and the 
r mi rf if 

required receiver -input RF signal-power level, P . The system 

design curves may also be used to estimate system performance when 

either or both B and P are specified. Furthermore, system per- 
rf r 

formance may be estimated for a time - varying median- value of P ; 
this time -varying median P is obtainable by methods outlined in 
section 7. 



-108- 
6.4 Test Procedures for SSBSC-AM and FM Systems 

Acceptance tests on radio communication systems involve the 
measurement of the system performance, in terms of the message 
error rate for particular message loads and transmitter output power. 
Test programs should be based on acceptable message-error rates, 
for a specified number of simultaneously-active message channels, 
with a given radio-frequency spectrum bandwidth, and for a particular 
value of transmitter -output power. The information contained in the 
system design curves of this report may be used as a guide for deter- 
mining proper test procedures, and also for estimating equipment test- 
level values for the system variables and parameters, as applied to 
SSBSC-AM and FM multi-channel radio communication systems. 
Acceptance-test programs may be arranged conveniently by following 
the system design procedures outlined in Sections 6.1, 6. Z, and 6.3. 
The major differences between design procedures and test procedures 
are due to the fact that in the former case certain parameters are to 
be estimated while in the latter case these same parameters are 
specified and the system performance is to be measured. In either 
case the design equations and the design curves in this report may be 
used to obtain quantitative estimates of the performance of the radio 
communication system. 

6.5 System Performance Estimates for 
SSBSC-AM and FM Systems. 

Using the procedures outlined in Sections 6.1, 6. Z, 6. 3, and the 
appropriate design curves and equations given in this report, the per- 
formance of a specific radio communication system may be estimated 

in terms of the received RF-signal power, P , and the corresponding 

r 

message error rate. One form of presentation of these results is 



-109- 
shown in figure 6-1. Note that the curves in this figure are only pre- 
sented as being typical and are not accurate, hence they do not apply 
to a particular system; the same is true of figures 6-2 and 6-3. 

A typical measured or estimated distribution [ Rice, Longley, 
Norton; 19^9] of a received tropospheric radio signal is shown in 
figure 6-2. The "sampling periods" should be restricted to intervals 
during which the distribution of the received RF signal approximates 
a Rayleigh distribution. The sampling period may also be defined in 
terms of an interval of time which is very large compared with the 
duration of an information bit; this definition permits the transmission 
of a "usable" sample of the message during the sampling period. The 
sampling-period median received-power level, P , is sometimes refer- 
red to as the "hourly median", with the (implied) understanding that it 
is not necessarily the median for an hour but that the period of time 
might only be a fraction of an hour, or it might be several hours; the 
above definitions of the sampling period may be used to clarify the 
meaning of the hourly median, of the received RF signal-power level. 
The total time involved in the percentage-of- sampling periods scale 
must be very much greater than the sampling-period time. 

Combining the data in figure 6-1 with that in figure 6-2, using the 
method outlined in Appendix D, we obtain the distribution of the mes- 
sage error rate as shown in figure 6-3. Note that the positions of 
these curves, relative to the message error-rate scale, depend 
directly upon the median power level, P , of the received RF- signal, 
as given in figure 6-1. The total time to be considered in figure 6-3 is 
identical to the total time indicated in figure 6-2; that is, if figure 6-2 
represents the distribution of the received RF-signal power levels 
for one day, then figure 6-3 will show the distribution of the message 
error rate for the same period of time, that is, for the same day; 
similar results may be obtained for a portion of a day, for one week, 
one month or for any specified total time. 



-110- 



MESSAGE ERROR RATE VERSUS RECEIVED RF-SIGNAL 




-105 



P r , NON -DIVERSITY SAMPLING-PERIOD MEDIAN 
OF RECEIVED RF-SIGNAL POWER, dbw 

Figure 6-1 



-Ill 



< 

Q 
UJ 

o-d 

a: . 

UJ _i 
Q_ < 

I 11 
t> 

— lil 
UJ a; 

i c 



DISTRIBUTION OF MEDIANS OF RECEIVED 
RF-SIGNAL POWER 



-115 



-120 



-125 



-130 



-135 I 



0.5 5 10 20 30 40 50 60 70 80 85 90 95 98 99 99.5 

PERCENTAGE OF SAMPLING PERIODS ORDINATE VALUES ARE EXCEEDED 

Figure 6-2 



112 



DISTRIBUTION OF MESSAGE ERROR RATE 



IU 












































































7 










































































5 










































































3 
2 

i 
































NON-DIVEF 


*SITY 












































































i 












































































0.7 


































" DUAl^^*" 






































0.5 










































































0.3 
0.2 

0.1 


































QUADRUPLE 




















































































































































0.07 










































































0.05 










































































0.03 
0.02 

001 















































































































0.01 I 5 10 20 304050 60 70 80 90 95 98 99 99.5 99.9 99.95 

PERCENTAGE OF SAMPLING PERIODS DURING WHICH THE MESSAGE 
ERROR RATE IS LESS THAN THE ORDINATE VALUE 



99.99 



Figure 6-3 



-113- 

The information in figure 6-3 is presented in a form which may be 
easily and unambiguously interpreted. The statistical nature of the 
message error rate is clearly indicated by these curves. From data 
of the type shown in figure 6-3 it is possible to predict the system 
performance for various periods of time such as, all-year, worst-day, 
etc. 

The data in figure 6-1 would apply to a particular message load on 

the specified system. The material in this report may also be used 

[Barsis, et al. , 196l] to estimate radio communication system 

performance for various assumed system-parameter values, and the 

results may then be presented in terms of the percent message error 

and the various parameters such as, the RF- signal spectrum bandwidth, 

B , number of message channels, N, percent message load, etc. This 
ri 

type of presentation of system performance estimates may be applied 

to specific cases, and it is sometimes more convenient to use this form 

of presentation than to use the general form of design curves shown in 

this report. However, these design curves are basic and of course are 

not restricted to specific cases. 

Test procedures involve the calibration of the receiver so that it 

may be used to indicate the power level of the received RF signal. The 

receiver-input signal power, P , is available at the receiver -input 

terminals. Hence, when calibrating a receiver with a signal generator, 

using impedance mismatch conditions at the receiver input, in order to 

simulate operating conditions, care must be exercised to make certain 

that the values of P obtained from the signal generator output-power 

r 

scale are properly corrected. These corrections would be in terms of 
the output impedance of the signal generator and the characteristics of 
the impedance -changing unit, which may be required between the 
generator output and the receiver input. Proper corrections (see Ap- 
pendix A) will yield the available RF signal power at the receiver input 
terminal. 



-114- 
7. REQUIRED TRANSMITTER-OUTPUT POWER 

Under conditions where the signal power is transmitted over a 
radio path, the required transmitter -output RF- signal power, P , 
may be calculated from the following (equivalent) equations: 



P = (P L T L, L L. L, )/G (7.1) 

t r T bm c t du p 



or, 



10Log 1() P t =10Log 1() P r + 10Log 10 L T + 10Log 1() L^ 

(7.1a) 

- 10Log,„ G + 10Log lrt (L L L n ), dbw 
5 10 p 5 10 c t du' 



where, 



P = Transmitter output average power, watts. 

P = Required available receiver -input median of the 
non-diversity signal power level, watts. Median 
values of P are used primarily because of the 
fact that calculations of the path transmission 
loss, L, , yield results in terms of median 
values for L» . Furthermore, it is easier to 
obtain measured values of median P (or L. ) 
from their (measured) distributions than it is to 
obtain average values for these factors. 

L = Total antenna- system transmission loss factor 
at the transmitter, as a ratio. 

L, = Median of the basic transmission loss. This 
bm 

factor is defined as the ratio of the power input 

to the terminals of a loss-free isotropic trans- 
mitting antenna to the power available from the 
terminals of a loss-free isotropic receiving 
antenna. The median basic transmission loss, 
L , and the effective path gain, G D , may be 
calculated by methods outlined in NBS Technical 
Note #15. [Rice, Norton, Longley, 19 59] and NBS 
Technical Note #101 [Rice, et al. , 1962] . 



-115- 
* 



' , L , = Transmission loss factors for the (receiving) antenna 
c t du . , , , , 



circuit, transmission line, and the duplexer network, 
respectively, as a ratio. These factors are defined 
as the ratio of the power delivered to the input of a 
network to that portion of the input power which is 
available at the network output. These factors are 
conveniently measured for non- reflecting impedance- 
matched conditions at the input and at the output of 
the network. 

G = Effective path gain [ Hartman, Wilkenson, 1959; 
JTAC, I960] of the transmitting and receiving 
antennas combined, relative to an isotropic antenna, 
as a ratio. 

Eq. (7. 1) may also be written [ Norton, 1956] as: 

P t = (R B. r F L L kT )/G (7.2) 

t if T bm o" p x 

or, 

10Log 1() P t = 10Log 1() R + 10Log i0 B. f + 10Log 1() F + 10Log 1() L T 

(7.2a) 

+ 10Log in L. - lOLog, _ G - 2 04, dbw 
°10 bm °10 p 

The factor F in (7.2) is the "effective noise figure," [ Barsis, 
Norton, Rice, Elder, 19&1, Appendix III] . For a receiver with no spur- 
ious responses, 

T + (L - 1) T + L (L - 1) T + L LJF - 1) 
a x c c c t t c t r , ? 3 > 

o 

The factors in (7. 3) are identical to the factors used in (4. 3) and 
(4.3a), provided that impedance -matched conditions exist in the receiv- 
ing antenna system; these factors are described in Appendix A. 



* These factors apply only to the case where non- reflecting impedance- 
matching conditions prevail in the transmitting and receiving antenna 
systems, except at the receiver input terminals. 



-116- 
The factor R in (7.2) is the predetection signal-to-noise ratio, 

P.yN.,.. The relationship between R and P maybe obtained by 

if if r 

combining (7.1) and (7. 2), and is 

RB.,FkT 

P = . 1 . ° (7.4) 

r L L. L n 

c t du 

Multiplying both sides of (7.4) by l/f (T + T ), we obtain 



P RB.JkT 

lf ° (7.5) 



f (T A + T ) L L L J f (T A + T ) 

mA er ctdumA er 

From (4.3) and (7.3), and neglecting the term P /k B L , we have 



L L. L, (T. + T ) 
F= C t du ^ A ^- (7.6) 



T 
o 



Combining (7. 5) and (7. 6) we obtain 



P R B k R B k 

r if rf 



f (T A + T ) f f 

m A er m m 



(7.7) 



Equation (7.7) is useful when it is desired to obtain values of the 
predetection signal-to-noise ratio, R, from the system design curves; 
this equation gives the appropriate substitution of abscissa-title factors 
in the design curves in figures 4-2, 4-4, 4-5, 4-6, and 4-7. 

The required transmitter -output power, P , may be determined in 
terms of either: 



-117- 

(1) the required receiver -input power level, P , which is obtainable 

r 

directly from the system design curves and then substituted in 
(7. 1) or (7. la), or 

(2) the required predetection signal-to-noise ratio, R and the required 

bandwidth, B (= B.J; values for these factors are inserted in 
rf if 

(7.2) or (7. 2a). 

The choice between the use of either of the above procedures, 

(1) or (2), depends entirely upon the designer's preference; however, 

where performance -test measurements are made it is very much 

more convenient to measure P than to measure R. Furthermore, 

r 

(7.1) and (7.1a) contain only power levels, loss factors and gain 
factors and hence these equations are easily understood. Equations 

(7.2) and (7. 2a) involve the additional factors, F and R, each as a 
ratio; this situation apparently is confusing to many engineers. 

The following relationships between P /N. f and the various 
system parameters have been found to be useful, and are included 
here for convenient reference: 



P. f P 

R = — — = (7.8) 

N. r k(T A + T ) B.. 

if A er if 



P 

a 



L L L k(T + T )B. r 
c t du A er if 



(7.9) 



P G 
t p 



L L, L L L, k(T A + T ) B. , 
T bm c t du v A er' if 



(7.10) 



where, 



-118- 

-23 

k = Boltzman's constant = 1.3804 x 10 

T = Effective receiving-antenna system output 

noise-power temperature, in degrees Kelvin; 
see Appendix A, pp. 157-161. 

T = Effective receiver-input temperature, in 
er 

degrees Kelvin. This factor is given by 

T = (F - 1) x 290. 
er r 

P = Received RF-signal, available at the terminals 

of the "loss-free" antenna. 



From (7.8), (7.9), and (7. 1 0) we have, 



P 

(7.11) 



r L L L, 
c t du 



P r ' L T U "I L L < 7 -> 2 > 

T bm c t du 



-119- 
CONCLUSIONS 

The equipment-performance design curves presented in this report 
are based on the performance of "ideal" radio receivers and maximal- 
ratio type of combiners and hence these curves will yield optimistic 
results. However, for FM systems, placing the single sinusoidal 
modulating signal in the highest (baseband) frequency partially compen- 
sates for the optimistic assumptions regarding receiver (ideal) per- 
formance and combiner operation. Furthermore, proper allowance 
may be made for the effect of system inter modulation noise using the 
methods outlined in subsection 4. 3. 6 and following the examples worked 
out in section 6. 

In practice, the degree of conformity between calculated and mea- 
sured system performance will depend largely upon the performance 
of the combiner, compared to the (ideal) performance of the maximal- 
ratio combiner. In fact, a comparison between the calculated and the 
measured performance of a system constitutes a reliable check on the 
combiner performance, after due allowance is made for all other factors 

The method used to obtain the equipment-performance curves 
implies that pre -detection diversity combining is used; the use of these 
results, where post-detection combining is employed, might be 
questioned. However, if the combiner operates so as to improve the 
distribution of the signal-to-noise ratio of the (combiner) input signals, 
then our methods are justified; since the overall results should be sim- 
ilar whether the combiner precedes or follows the receiver demodu- 
lator. 

The results of this work are presented in a form such as to be 
directly usable in determining system performance when combined with 
effective antenna gain and radio -path transmission-loss estimates. 



-120- 

The design curves shown are basic and may be used to determine 
sets of curves showing the system-performance characteristics for 
various combinations of the system parameters. These design curves 
involve the fundamental parameters of the system and hence may be 
used to establish equipment-performance standards on a quantitative 
basis. 

It should be understood that the technical analysis of FM and 
SSBSC radio communication systems carried out in this paper does 
not involve economic factors or operational complexities. However, 
this type of technical analysis is required as a starting point for the 
determination of a complete analysis and comparison of the different 
types of systems. 

In conclusion, a method has been outlined in this report, and a 
set of system-design curves have been developed based on the com- 
bined performance characteristics of the receiver, diversity combiner, 
modulator, and the message -signal decoder unit. The net results of 
this work are a set of curves in which all of the important system 
parameters are used as scale factors. The technique employed in 
this work essentially provides a link between information theory results 
and the application of these results. 






-121 

ACKNOWLEDGE ME NTS 

The material for this report was obtained through the assistance of 
a large number of individuals. Of these, the more important are: 
M. J. Ogas, who assisted in obtaining financial support from the USAF; 
Colonel Wm. E. Geyser, USAF, for permitting us to make use of the 
"Pole -Vault" tropospheric radio link between Gander, N. F. , and 
Pepperrell AFB, St. Johns, N. F. ; and Mr. Carlton J. Modlin, who 
made a major contribution to the experimental work carried out on the 
tropo system at Verona, N. Y. We are indebted to Mr. R. W. Brauer 
of the Comm. Sys. Eng. Div. , USASEA, for providing the tropo link 
between the East Coast Relay Station, Frederick, Md. , and La Plata, 
Md. , to carry out system-performance tests; and also the funding of 
the project. The cooperation of the Commanding Officers at the ECRS 
and the La Plata Station, Major Alfred K. Granschow and Major Howard 
L. Hall, and the assistance of the personnel at these Radio Comm. 
Stations, is gratefully acknowledged. Other persons who were of 
assistance in this program are: Sgt. George H. Farmer, R. F. 
Kaltenbach, and Sen. Sgt. R. L. Prather, all of Pepperrell AFB; 
Richard V. Locke, Burt E. Nichols, and Dave Karp of Lincoln Labo- 
ratories; Joe S. Turner, Wm. Long, L. R. Goodell, and Keith Garlets 
of Collins Radio Co.; Don Glen, B. F. Quereau, J. A. Clark, B. D. 
Samsel, and H. R. Dahms of the National Bureau of Standards, Boulder 
Laboratories. Mr. John C. Harman and his assistants in the Drafting 
Department of the Boulder Laboratories are to be commended for the 
accuracy of the drawings. Mrs. Marylyn Olson, Mrs. Doris Hunt, 
and Mrs. Ruth H. Rotherham deserve credit for the tedious job of 
typing this report. 

For constructive criticism of this report, we are grateful to: 
K.A.Norton, R.C.Kirby, C. Gordon Little, A.P.Barsis, W.O. 
Crichlow, R.W.Beatty, W. Beery, P. Hudson, C. Allred, and many 
others. 



-122- 

REFERENCES 

Albersheim, W. J., and Schafter, J. P., Echo distortion in the FM 
transmission of frequency-division multiplex, Proc. IRE 40 , 
316-328 (March 1952). ~~ 

Barsis, A. P., Norton, K. A., Rice, P. L. , and Elder, P.H., Per- 
formance predictions for single tropospheric communications 
links and for several links in tandem, Tech. Note #102, National 
Bureau of Standards, Boulder Laboratories (Aug. 1961). 

Becking, A. G. Th. , Groendijk, H. , and Knol, K.S., The noise 

factor of four -terminal networks, Philips Res. Rept. 10, 349- 
357 (1955). 

Bell, R. L. , Induced grid noise and noise factor, Proc. IRE (Sept. 1951). 

Beranek, L. L. , The design of speech communications systems, Proc. 
IRE, 35, 880 (1947) 

Black, Harold S. , Modulation theory, p. 224 (D. Van Nostrand and 
Co., 1953). 

Black, HaroldS., Modulation theory, p. 221 (D. Van Nostrand and 
Co., 1953) 

Brennan, D. G. , Noise methods II: Linear diversity techniques, 

Lincoln Lab. Group Report 36-29, p. 56, p. 59 (Mar. 1, 1958). 

Carson, J. R. , Notes on the theory of modulation, Proc. IRE, 10, 57 
(1922); Proc. IRE 7, 187(1929). 

Clutts, C. E., R. N. Kennedy, and J. M. Trecker, Results of band- 
width tests on the 185-mile Florida-Cuba tropospheric scatter 
system (I960). 

CCIR Report No. 65, Revision of atmospheric radio noise data, Inter- 
national Telecommunication Union, Geneva (1957). 

Crichlow, W. Q. , D. F. Smith, R. N. Morton, and W. R. Corliss, 
Worldwide radio noise levels expected in the frequency band 
10 kilocycles to 100 megacycles, NBS Circular 557 (Aug. 25, 1955). 

Eagan, J. P., Articulation testing methods II, U.S. Dept. of Commerce 
Report No. PB22848 (1944). 






-123- 

Ewen, Harold I., A thermodynamic analysis of Maser systems, Micro- 
wave Journal, 41-46 (March 1959). 

French, N. R. , and J.C. Steinberg, Intelligibility of speech sounds, 
J. Acous. Soc. of Amer. _19, 90 (1947). 

Friis, H.T., Noise figure of radio receivers, Proc. IRE, 419-422 
(July 1944). 

Gladwin, A.S. , Energy distribution in the spectrum of a frequency 
modulated wave, Part II, Phil. Mag., 38^ 229-251 (April 1947). 

Grimm, H.H., IRE Trans, on Instru. , 97-103 (Dec. 1959). 

Grimm, H.H., Noise temperature in passive circuits, Microwave 
Jl., 52-53 (Feb. I960). 

Hansen, R.C., Low noise antennas, Microwave Jl. , 19-24 (June 1959). 

Harris, Donald P. , An expanded theory for signal to noise performance 
for FM systems carrying frequency division multiplex, IRE Con- 
vention Record, Part 8, 298 (1958). 

Hartley, R. V. L, Transmission of information, B.S.T.J., vol. 7, 
July 1928, 535-563. 

Hartman, W.J. and R.E. Wilkerson, Path antenna gain in an expo- 
nential atmosphere, Jour, of Res. of the Nat. Bur. of Stda. -D, 
Nov. -Dec, 1959. 

Haus, A.H. , and R. B. Adler, Circuit theory of linear noisy networks, 
Tech. Press of MIT and John Wiley & Sons, 1959. 

Haus, A.H. , and R. B. Adler, Optimum noise performance of linear 
amplifiers, Proc. IRE, Aug. 1958. 

Hogg, D.C. and W.W. Mumford, The effective noise temperature of 
the sky, Microwave Jl. , 80-84 (March I960). 

Hund, August, Frequency modulation, first ed. , 33 (McGraw-Hill 
Book Co. , Inc. , 1942). 

IRE Committee on Noise, Standards on electronic devices: Methods of 

measuring noise, Proc. IRE, 891-896 (July 1953); 60-68 (Jan. I960), 

IRE Committee 7. 9 on Noise, Representation of noise in linear twoports, 
Proc. IRE, 69-74 (Jan. I960). 

ITT Handbook, Reference data for radio engineers, Fourth Edition 
(1956). 

Joint Technical Advisory Comm. , Radio transmission by ionospheric 
and tropospheric scatter, Proc. IRE, (Jan. I960). 



-124- 

Landon, V.D., Theoretical analysis of multiplex transmission, RCA 
Review (June, Sept. 1948). 

Medhurst, R. G. , RF bandwidth of frequency-division multiplex using 
frequency modulation, Proc. IRE, 189-199 (Feb. 1956). 

Middleton, D. , The distribution of energy in randomly modulated waves, 
Philosophical Mag. , 42, 689-707 (July, 1951). 

Norton, K.A., Point-to-point radio relaying via the scatter mode of 

tropospheric propagation, IRE Trans, on Communications Systems 
CS-4, No. 1, 39-49 (1956). 

Nyquist, H. , Certain factors affecting telegraph speed, B.S.T.J., vol. 
3, Apr. 1924, pp. 324-326. 

Nyquist, H. , Certain topics in telegraph transmission theory, AIEE 
Trans., vol. 47, Apr. 1928, pp. 617-644. 

Nyquist, H. , Thermal agitation of electric charge in conductors, Phys. 
Rev. 32, 110 (July 1928). 

Plush, R. W., A. D. Watt, and E. F. Florman, Private communication 
(I960). 

Rice, P. L. , A.G. Longley and K. A. Norton, Prediction of the cumu- 
lative distribution with time of ground wave and tropospheric wave 
transmission loss, NBS Tech. Note #15 (July 1959). Available 
at a cost of $1.50 from the Office of Technical Services, U. S. 
Department of Commerce, Washington, D.C. Foreign remit- 
tances must be in U.S. exchange and must include one-fourth of 
the publication price to cover mailing costs. 

Rice, P. L. , A. G. Longley, and K. A. Norton, Transmission loss 
predicition for tropospheric communication circuits, N.B.S. 
Technical Note #101. 

Rothe and Dahlke, Theory of noisy fourpoles, Proc. IRE, 811-818, 
(June 1956). 

Siegman, A.E. , Thermal noise in microwave systems, Part I, 
Microwave Journal, 81 (March 1961). 

Siegman, A.E. , Thermal noise in microwave systems, Part II, 
Microwave Journal, 66-73 (April 1961). 

Slack, M. , The probability distributions of sinusoidal oscillations 
combined in random phase, J. IEE 93, Part III, 76-78 (March 
1946). 

Staras, Harold, Forward scattering of radio waves by anisotropic 
turbulence, Proc. IRE, 1380 (Oct. 1955). 



-125- 

Stewart, J. L. , The power spectrum of a carrier frequency-modu- 
lated by Gaussian noise, Proc. IRE, 1539-1542 (Oct. 1954). 

Stumpers, F.L.H.M., Theory of frequency modulation noise, Proc. 
IRE, 36, 1081-1992 (Sept. 1948). 

Strum, P.D., A note on noise temperature, IRE Trans. MIT-4, 145, 
(July 1956). 

Terman, F.E., Radio Engineers Handbook, First Edition, 172-210, 
(1943). 

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Wait, James R. , Transmission of power in radio propagation, Elec- 
tronic and Radio Engineer, April, 1959. 



-126- 

LIST OF SYMBOLS 



B Composite baseband signal spectrum bandwidth, c/s. 

b 

B Message -signal spectrum bandwidth in c/s. In a single - 

sideband frequency-division multiplex system B is also 
the bandwidth occupied by the message signal in the com- 
posite modulating baseband signal. 

B.. Effective (noise-power) bandwidth of the receiver IF 

circuit, at the receiver demodulator input. 

B Effective bandwidth of a network, in c/s. 

n 

B f Radio-frequency signal power -spectrum bandwidth of 

the total radio-frequency signal, in c/s. 

B - Radio-frequency signal FM spectrum bandwidth for a 

white-noise type of modulating signal. 

B Radio-frequency signal FM spectrum bandwidth for a 

sinusoidal modulating signal. 

D Modulation-demodulation conversion factor. This 

s s 

factor is defined as, P (100%)/P , and applies 

only to amplitude modulation. 

F Effective noise figure (Barsis, et al, Appendix III, I960). 

f , f , f Frequency in c/s of particular signals in the modulating 

composite baseband signals. 

f Maximum value of the modulating- signal frequency in 

c/s. Also, the highest frequency in a composite base- 
band modulating signal. 

F Noise figure of a network, as a ratio; see figure A-l. 

F Noise figure of (radio) receiver, or amplifier, as a 

r 

ratio, measured according to IRE Standards (IRE Comm. 

1953, I960). 



-127- 



G r Power gain of the linear section of the (radio) receiver, 

or amplifier. This factor is defined as the ratio of the 
available output power to the power delivered to the 
receiver input. This is not the available gain. 






-L 3 
k Boltzman's constant = 1.38042 x 10 joules per degree 

Kelvin. 

K Bandwidth- compression factor, applied to the receiver. 

This factor indicates the degree of IF bandwidth com- 
pression in the receiver; note that < K < 1. 

c 

L Transmission loss factor for the antenna network, due 

c 

to the lossy elements in the receiving antenna. This 

factor is equal to the ratio of the non- reflecting impe- 
dance-matched delivered power at the network input to 
the non-reflecting impedance-matched power at the 
output of the network. 

L Ditto, for the duplexer. 

L Ditto, for the antenna- system transmission line. 

L Transmission loss factor of a network; defined as the 

ratio of the non- reflecting impedance-matched input 
power to the non- reflecting matched-impedance output 
power. 

L. Isolation factor for the duplexer. This factor is equal 

to the ratio P« f /N. , see figure 4-1. 

L Total transmitting antenna system transmission loss, 

as a ratio. 

m Amplitude -modulation index, in percentage. 

M. Impedance mismatch factor, at network input, 

in 

M Impedance mismatch factor at the receiver input term- 

lr \ 

mals. 

M Number of simultaneously- active message signals each 

of which requires the same value for the product 

< B c< S o</ N oc> >A- 

M T Ditto, for the product (B (S /N ) ) T . 

I c oc oc I 



-128- 

M Ditto, for the product (B (S /N ) ) . 

N c oc oc N 

N Receiving-antenna environmental noise power, in watts. 

3 

N Ditto, for interfering signals. 

3.1 

N Available noise power at the output terminals of a net- 

ao , r- a i 

work, in watts; see iigure A-l. 

N Receiving antenna noise power, transmitted with the 

RF signal, in watts. 

N Available noise power at receiving-antenna system out- 

put, due to antenna temperature, T , plus noise gener- 

3. 

ated within the antenna- system units. 

N Noise power generated within and contributed by a net- 

work, in watts. This noise power is referred to the 
input terminals of the network, see figure A-l. 

N Total of the average noise power at the IF output, or at 

the demodulator input, in watts. 

N Receiver output average total noise power, in watts. 

N Average noise-power level of a message-signal chan- 

nel having an effective bandwidth of B , at the receiver 
output; or, a relative average noise -power level per 

bandwidth, B , at decoder-unit input terminals, 
c 

N Total available noise power from the receiving-antenna system 

p Average power level of a message-signal in the modu- 



m 



mi 



lating baseband signal, in watts 



1"U 

p Average power level of the i — message signal in the 



modulating baseband signal, in watts. 



P Receiver IF-output total average signal power, or, 

average signal power at the demodulator input, in watts 

P Power output of the local transmitter, in watts. 






-129- 

P Total average power level of the baseband modulating 

signal. It is understood that this power level is deter- 
mined under conditions when a specified number of 
simultaneously-active message signals are in operation. 

P 

Message loading ratio, being the ratio of the total 

m average power level of the modulating signal to the 

average power level of a message signal. 

P Average power level of a "white -noise" modulating 

signal, in watts; where the modulating signal is distribu- 
ted uniformly across a specified portion of the modulat- 
ing baseband. 

P Peak-power level in the composite modulating signal, 

' exceeded for only a specified percentage of the time. 

P Average power level of a sinusoidal modulating (system 

test) signal, in watts; under conditions where only one 
modulating signal is used. For FM systems the fre- 
quency of this modulating signal is f --the highest 
frequency in the baseband signal. 

P (100%) Sinusoidal- signal power level required to produce 100% 

amplitude modulation. 

P Total average radio-frequency signal power available at 

the receiver -input terminals, in watts. For a time-vary- 
ing RF signal, the median value of P is used in this 
report. 

P Transmitter average total power output, in watts. 

P Peak-power level at the transmitter output, exceeded for 



tp 



P 
N 



only a specified percentage of the time. 



R= T7 ^— Predetection total signal-to-total noise ratio. For the 

if case of a varying received signal, R will be in terms of 

its median value. 



S Receiver output (baseband) average total signal power, 

o 

in watts. 



-130- 

S Average message-signal power in the receiver output, or 

a relative message-channel signal average power level 
at the decoder -unit input terminals. 

T Receiving-antenna noise (power) temperature, associated 

with the sky noise and the environmental noise, in degrees 
Kelvin. This noise temperature, or its equivalent noise 
power N , is available at the output terminals of the loss- 
free antenna. 

T Ditto, for interfering- signal power. 

T Ditto, for the transmitter -output noise power transmitted 

with the carrier signal. 

T Thermal temperature of the lossy elements in the (receiv- 

er 

ing) antenna-circuit network; see figures A-l, 4-1, and 
Appendix A. 

T Ditto, for the duplexer in receiving-antenna system. 

T Effective input noise (power) temperature of a network, 



ei 



er 



n 



o 



degrees Kelvin; see figure A-l 



T Effective output noise-power temperature of a network, 

degrees Kelvin; see figure A-l. 

T Effective receiver-input temperature, degrees Kelvin. 



(IRE Comm. on Noise, I960). 

T Output noise-power temperature of source or signal 

° generator, degrees Kelvin; see figure A-l. 

T Thermal temperature of the receiving-antenna transmis- 

sion line, degrees Kelvin. 

T Thermal temperature of the lossy elements in a passive 



network, degrees Kelvin; see figure A-l 



T Standard reference temperature, assumed to be 290. 



degrees Kelvin. (Ire Comm. on Noise, I960). 



T Effective output noise (power) temperature of the 

receiving-antenna system, degrees Kelvin. T is 
defined by: N = kT B , see Appendix A. 

1 J\ IX 



-131- 
z du Characteristic output impedance of the duplexer. In 

general this is the characteristic output impedance of 
the network which precedes the receiver. 

z ^ n Input impedance of network or amplifier, in ohms. 

z - Input impedance of the radio receiver. 

z Output impedance of source or signal generator, in ohms 

z on Output impedance of network or amplifier, in ohms. 

z j. Image or characteristic input impedance of a network 

or transducer. 



Z Jri Image or characteristic output impedance of a network 

or transducer. 



Io 



AF Peak radio-frequency carrier deviation above and below 

the un-modulated carrier frequency, in c/s; for a 
single sinusoidal modulating signal having an average 
power level P , equal to the total average power level 
of the composite modulating signal. 



AF^ Peak radio-frequency carrier deviation in c/s; due to 

the message-signal modulating power, p 



c 



m 



AF 

— — Deviation ratio, in radians. This ratio is equal to the 

"phase deviation" of the radio-frequency carrier, that 

is, the extreme angular displacement of the carrier 

from its "average" position. 

P Correlation (radiofrequency) bandwidth in Mc/s. This 

bandwidth is defined as the frequency separation in c/s 
of spectral components whose cross-correlation coef- 
ficient has an average value of 0. 5. 

c|>(AF/f) Calculated ratio of the radio-frequency spectrum band- 

width B -, to the deviating frequency f ; applicable only 
to the case where a single sinusoidal modulating signal 
is used--see figure 3-1. 



-132- 

Figure No. Page No. 

2-1 Block Diagram of Basic Units in a Multi- 

channel Tropospheric Radio Communication 
System 6 

3-1 Relationship Between the Deviation Ratio 

AF/f m and RF Spectrum Bandwidth, B f , 
which Includes 99. 99 Percent of the RF 
Spectral Power 12 



3-2 



Receiver Output S /N Ratio Versus B e /V>- c 

oc oc rf if 

for Various Receiver -Input Total RF Signal 

Power Levels Steady Receiver-Input Signal 

Sinusoidal Modulating Signal 14 

3-3 Relative Levels of A F and P^,, Sinusoidal 

m 

and Noise Modulating Signals, and for Equal 

RF Spectrum Bandwidths 19 

3-4 RF Spectrum Bandwidth, B f , Versus Rela- 

tive Modulating-Signal Power Levels, for 
Frequency Modulation 2 

3-5 Amplitude Modulation Exceeding 100 Percent 

Versus Relative Modulating-Signal Average 
Power Levels, For SSBSC-AM 26 

4-1 Block Diagram of Receiving System 136 

4-A FM Receiver-Output Signal and Noise in the 

Top Channels of the Baseband Signal 42 

4-2 SSBSC-AM Radio Receiver Performance 

Curves Steady and Rayleigh-Fading Types 
of Received Signal 64 

4-3 FM Radio Receiver Characteristic Curves 

Steady Received Signal 72 

4-4 FM Radio Receiver Performance Curves 

Steady Signal 76 



Figure No, 
4-5 

4-6 
4-7 

4-8 



4-9 



-133- 



FM Radio Receiver Performance Curves 
Rayleigh-Fading Signal, Non-Diversity 

FM Radio Receiver Performance Curves 
Rayleigh-Fading Signal, Dual Diversity 

FM Radio Receiver Performance Curves 
Rayleigh-Fading Signal, Quadruple 
Diversity 

FM Receiver Characteristic Curves, 
Without Pre-Emphasis 

Effect of Pre-Emphasis on FM Receiver 
Characteristics Curves 



4-10 Effect of Bandwidth Compression on FM 

Receiver Characteristics 

4-11 Effect of Bandwidth Compression on FM 

System Design Curves 

5-1 Cumulative Amplitude Distribution of Noise 

at Receiver Output for Received Carrier 
Power Levels Above and Below the FM 
Radio Receiver Threshold Level 



Page No, 



78 



79 



80 



82 



84 



85 



87 



89 



5-2 Measured Relationship Between the Steady- 

Signal Power to Mean Noise Power Ratio, 
at the Teletype Receiver Input Terminals, 
and the Average Teletype Character Error 
Rate 

5-3 Measured and Calculated Relationship 

Between the Signal Power to Mean Noise 
Power Ratio, at the Teletype Receiver 
Input Terminals, and the Average Character 
Error Rate at the Teletype Printer Output 



90 



92 



-134- 

Figure No. Page No, 

6-1 Message Error Rate Versus Received RF 

Signal 110 

6-2 Distribution of Medians of Received RF 

Signal Power 111 

6-3 Distribution of Message Error Rate 112 

A-l Network Noise Powers and Noise Temper- 

atures 145 

D-l Graphical Method for Determining FM 

Receiver Performance 177 

D-la FM Receiver Performance Characteristic- 
Steady Received Signal 177 

D-lb Cumulative Distribution of Maximal-Ratio 
Combiner Output, for Dual and Quadruple 
Diversity, for Rayleigh-Distributed Input Signal 177 

D-lc Distribution of FM-Receiver Output Signal-to- 

Noise Ratio-Time Varying Received Signal 177 

D-ld FM Receiver Performance Characteristic- 

Rayleigh-Fading Signal 177 

D-2 Graphical Method for Determining Message- 

Decoder Performance 181 

D-2a Performance Characteristic of Teletype 

Decoder Unit-Steady Signal-to-Noise Ratio 181 

D-2b Distribution of FM-Receiver Output Signal-to- 
Noise Ratio-Time Varying Received Signal 181 

D-2c Distribution of Teletype Character Error 

Rate for Time-Varying Received Signal 181 

D-2d Performance Characteristic of Teletype 
Decoder-Unit for Time-Varying Received 
Signal 181 



-135- 

APPENDIX A 

Pre-Detection Noise-Power Levels in the Receiving System. 

The objective is to obtain the functional relationship between the 

predetection noise-power level, N in figure 4-1, and the various 

receiving-system parameters: F , L , G , T , etc., as indicated 

x x x x 

in this figure. The method used to determine the total pre-detection 
noise-power level in the receiver consists of properly adding the noise- 
power outputs of the receiving-system (linear ) twoport network 
units in cascade [ Ewen, 1959; Hansen, 1959] . Basically, the noise 
performance of each network is considered in terms of the amount 
of noise contributed by the network, and the resultant effect of this 
noise on the network- output signal-to-noise ratio. 

In this analysis, the noise performance of each twoport network 
unit in the receiving system is determined in terms of the total noise 
power delivered* to and absorbed at the input of the unit (including the 
noise power originating in the unit) and the resultant available* noise 
power at the output of the unit. The use of delivered noise powers at 
the input of the units results in noise-performance factors which de- 
pend primarily upon the internal characteristics of the network 
units, such as power gain, effective bandwidth, image or character- 
istic impedance, and transmission loss . Furthermore, network 
noise figures, and noise performance characteristics, may be defined 
in terms of delivered (signal and noise) powers instead of available 
powers at the network input, without any loss in generality. The 
use of the delivered-power concept has the additional advantage of 



*If impedance mismatch conditions exist at the input terminals of the 
network, the delivered power is dependent upon the input impedance of 
the network and the output impedance of the preceding network. The 
available power is the maximum power which may be obtained from the 
network, at its output terminals; see following subsection. 



136- 



LU 

co 

>• 
co 

CD 

2 

> 

q: 



< 

o 
< 

Q 

o 
O 

_i 

CD 




i 

• 



-137- 

making it possible to define and to calculate the impedance mismatch 

factors in terms of the various system impedances, independently of 
the power gain or the dissipative transmission loss, and the effective 
bandwidth of the networks. 

The available -power concept, as applied to the noise power (and 
also the signal power) at the network output terminals, is valid 
because we are restricting this analysis to receiving antenna- system 
units having positive input and output resistances. In other words, the 
"exchangeable-power" concept is not required in this case [Haus, 
Adler, 1959] . 

The various factors discussed above such as delivered power, 
available power, transmission loss, etc. are defined and discussed 
in the following subsections. 

Effects of Impedance Matching on 
Delivered, Matched-Impedance, and Available Powers. 

In this appendix the effects of impedance mismatching is treated 
in a general manner, for the purpose of indicating how the impedance 
mismatch factors affect the noise-power level, or the noise tempera- 
ture, at the receiving-antenna system output terminals. No attempt 
is made here to derive the mismatch factors as explicit functions of 
the antenna system impedances. 

The noise performance properties of a linear lossy passive bi- 
lateral twoport network, of the type generally used in a receiving- 
antenna system, are dependent upon impedance -matching conditions 
at the network input and output terminals. The effects of impedance 
matching on network-input and output powers are determined from 
the following considerations, based on information in the Radio Engine- 
ers Handbook [ Term an, 1943] and the ITT Handbook [pp. 549-616] . 



-138- 
The power delivered to and absorbed by a load having an input 

impedance Z ( = R + j X ), when connected to the output terminals of 

a source or generator having an internal impedance Z ( = R + jX ), 

g g g 

and an internal (generated) voltage, E , is 

2 l E I 

P del = I * I R i = ', ^TT X R i (A - l) 

Z + Z I * 

1 g i ' 

where, I = Current flowing in the load. 

If the generator internal (or output) impedance is non-reflecting 
matched to the load impedance, that is, Z = Z , the power in the load 

becomes 

i i 2 

I E I 

P . = L- x R (A. 2) 

mat . l „ | c. a 

4 Z K 

g 



Note that the impedance-matched power, P , is absorbed in the 

mat 

load under non- reflecting impedance-matched conditions at the input 

terminals of the load. 

The maximum power will be obtained from the generator when the 

generator impedance is conjugate to the load impedance; that is, 

Z = Z„ , and hence R = R. , and X = -X„ . The available power is 

g * g * g i F 

defined as the maximum power which may be obtained from a given 

source, and is 

|E | Z |E | Z 

P s ^- x R. = S — (A. 3) 

aV 4R Z l 4R 

g g 

A study of (A. 1), (A.Z), and (A. 3) shows that: 

(a) the available power closely approximates the non- reflecting 

matched-impedance power only when the ratio X /R is very much 

g g 
less than unity; that is, X /r < <i } anc } 



-139- 

(b) conjugate -impedance matching is a special case of impedance 

(mis)matching and hence its use results in power reflections at the 
conjugate -impedance -matched point. 

The impedance mismatch factor is defined as the ratio of the 
power delivered to the load without matching to the power delivered 
with either type of impedance -matched conditions. Thus, the imped- 
ance mismatch factor may be defined in terms of either the non-reflect- 
ing matched-impedance power or the conjugate -impedance matched 
(available) power. In terms of the non- reflecting matched-impedance 
power, the impedance -mismatch factor is defined as 

P , _ 4 | Z |V 4(R 2 + X 2 ) R 
M s del_ = g_i = g gi {A4) 

mat P mat |Z +Z.| R | Z + Z J 2 R 

g | g g I g 

In terms of conjugate -impedance matched or available power, the 
impedance -mismatch factor is defined as 

P, , 4 R R„ 

- del g *■ ik c\ 

M =— = s (A. 5) 

av P i „ „ i c 

av Z + Z J 

g ^ 

From (A. 4) and (A. 5) it is evident that the impedance-mismatch 

factors for matched-impedance power and for available power are 

identical only when the ratio X /R is equal to zero. Hence, these 

~ g g 

powers are approximately equal when the ratio X /R is very much 

less than 1. In other words, maximum power transfer and non-reflect- 
ing impedance -matching can be attained simultaneously only when 

X /R < < 1 . 

g g 

When a network is inserted between the generator and the load, 

the impedance-mismatch factors at the network input and output term- 
inals may be determined in terms of the net or effective input and output 

impedances, Z and Z . In turn, Z. and Z may be calculated 
^ in on in on 



-140- 
from the network input and output image impedances, Z and Z , 

and the generator and load impedances, Z and Z [Terman, 1943] . 
The input image impedance of a network is defined as the impedance 
looking into the input terminals of the network with the output term- 
inals terminated by the output image impedance; note that the input 
and output terminals may be interchanged in this definition. The 
image impedance of a network may be determined from 

Z T = \T~Z Z =R T + iX T (A. 6) 

I oc sc I J I v 

where Z = Network input impedance with the network output 

oc : , M , 

terminals open-circuited 

Z ^ = Network input impedance with the network output 
terminals short-circuited. 

In a symmetrical network such as a transmission line, the input 
and output image impedances are equal to each other and also equal to 
the characteristic impedance of the line. 

The image impedances of a network may also be calculated in 
terms of the equivalent T or it network impedances [Terman, 1943, 
pp. 208-210] . Conversely, the equivalent T or it network impedances 
may be calculated for known or assumed values of network image im- 
pedances. These latter relationships are useful where it is desired to 

calculate the net or effective input and output impedances, Z. and Z , 

in on 

of a network in a tandem-network system in terms of network impedances 

In a system of tandem networks, the impedance mismatch factors, 

M. , at the input terminals of the networks are functions of the various 
in 

network image impedances, or the equivalent T or tt network imped- 
ances, and the generator and the load impedances. Each point or 
junction involving impedance mismatching may be treated as being 
equivalent to the simple case of impedance mismatch between a source 
(or generator) and a load; Z being replaced by Z and Z being 



-141- 

replaced by Z. . The network-input impedance mismatch factors may 
in 

be calculated by means of either (A. 4) or (A. 5); however, as stated 
previously, no attempt is made in this Appendix to derive the network- 
input mismatch factors explicitly in terms of the various impedances 
of the system. 

The results of the above derivations are to be applied to the anal- 
ysis of the noise performance of linear lossy passive bilateral networks 
in the antenna system; see pp. 147-151. 

Note that if non-reflecting (image or characteristic) impedance 
matching conditions are assumed to exist at all points in the system 
except at the receiver input terminals, and if the ratios, X / R , are 
assumed to be very much less than 1, throughout the antenna system, 
under these conditions the available power will be approximately 
equal to the matched-impedance power. Furthermore, at all of the 
impedance -matched impedance points the available power will be 
approximately equal to the delivered power. However, in the general 
case, where impedance mismatched conditions exist at more than one 
point in the system, and the ratios, X /R , are not each very much 
less than 1, it is possible to define and use mismatch factors (to 
estimate delivered power) in terms of either the non- reflecting match- 
ed-impedance power or the available power. 

Noise Performance of a Transducer Network 

Each lossy (bilateral) transducer unit, or network, in a receiving- 
antenna system contributes (average) available noise power which is 

equal to (F -l)kT B [ Friis, 1944] ; where F is the noise figure of 
non n 

the transducer unit, k is Boltzman's constant, and T is the "standard 

o 

reference temperature" of the generator impedance used to measure 

F and is assumed to be 290 degrees Kelvin [ IRE Comm. on Noise, 
n 

I960] . If the generator or source-impedance temperature, T , is 



-142- 

not T degrees, the correct of "standard temperature" value of F is 
° T T 



F = (1 - — °- ) + -=-°- F' ; where F' is the measured value of the noise 
n T T n n 

o o 

figure for the condition that T is not equal to T [ IRE Committee on 

* g H o 

Noise, I960, p. 64, eq. (6. a)] . 

The available noise power contributed by the network, N ., (see 

ei 

figure A-l) is referred to the input of the unit, and is added to the 

noise power available from the preceding unit, N , to give a total 

available noise power, N . + N . The "available -power transmission 

•ei ag 

loss factor" is defined here as 

N . + N 
_ ei ag . . , . 

ao 

where, N is the available noise power at the network output termin- 
ao 

als. 

In terms of delivered noise power at the network input and avail- 
able noise power at the network output, the "delivered power trans- 
mission loss factor" is defined here as, 

M. ) (N . + N ) 

^ ' ln) n " ag < A ^> 

ao 

where f M. ) is the network-input impedance -mismatch factor, in 
\ my av 

terms of available powers, and may be obtained from (A. 5). 
From (A. 6a) and (A. 6b) we obtain, 

l =/Sk < a - 6c > 

an /M. \ 

av 



in /M. N 

( in > 



For Impedance-matched conditions at the network input, M. = 1 

in 

and X =0 we have, 
g 

L, =L (A.6d) 

an n 



-143- 

Hence, the transmission- loss factor, L , may be considered as a 



n 



special value of L or L. , and is seen to be an internal character- 

an dn 

istic of the network. 

Note that in practice L, may be measured conveniently by using 

non- reflecting impedance -matched conditions at the input and output 

terminals of the network with the (test) signal-power-level, P , 

much greater than the transducer noise -power level, (F - l)kT B . 

Note also, that L is the "insertion loss" factor as defined by IRE 
n 

Standards [59 IRE 2. Si] . 

The transmission loss factor, L , is dependent upon the "real" 
part of the complex network "image transfer constant", Q, and is a 
measure of the dissipative loss of power in the network [ Terman, 
1943] . The image transfer constant, Q is given by 




tanh e = -x / ^ — (A. 7) 



Where transmission lines are involved, L , in decibels is pro- 

n 

portional to the product of the length of the transmission line, S. , and 
the "real" part (q; ) of the complex propagation constant, y; where 

Y = a + j£ . That is, 

L cc e 2Qi (A. 7a) 

n 

Network Bandwidth and Temperature Effects on Noise Power 

The noise power level at the output of a network depends upon 

the effective bandwidth of the network, B . The power accepted by 

n 

and processed in the network is proportional to B . Hence in a 
cascaded system of twoport networks, each of which may have 
different effective bandwidths, the network with the narrowest 
bandwidth will be the power-level and bandwidth-controlling unit. 



-144- 

It can be shown [ Siegman, 196la] that the exact expression for the 

(maximum or available) noise power originating in a lossy passive 

network having a thermal temperature of T degrees and an effective 

bandwidth, B , is given as, 
n 

h f B 

N n= exp(hf/kT)-l (A - ?b) 

where, h = Planck's constant 

f = Frequency 

hf 
since < < 1 even at microwave frequencies and normal tempera- 

tures, exp (hf /kT) ~ 1 + — r^ and (A. 7b) becomes 

iC JL 

N =kTB (A. 7c) 

n n 

and is valid for the radio frequencies and ambient temperatures used in 
radio communication systems. 

The effective bandwidth of the network, in c/s, is defined as: 

B n 5L ;IiV d£ (A - 7e) 

where L' is the effective network transmission-loss factor; see dis- 
n 

cussion of the various types of L' above, and L(f) is the network trans- 

n 

mission-loss factor, as a function of the frequency, f . 



Noise-Power Levels in a Network 

Noise power levels, and "effective noise temperature" [Strum, 
1956, 1958] , [Grimm, 1959, I960] , [Siegman, 196la, 196lb] , [ IRE 
Committee on Noise, I960] , at the input and output terminals of a 
linear passive lossy bilateral twoport network unit are shown dia- 
gramatically in figure A-l. Figure A-l applies to the case of im- 
pedance-mismatching conditions at the network input and output 



-145- 

terminals. The interrelationships between: noise figures, F , the 

respective effective input noise powers of the network and receiver, 

N . and N , and the corresponding effective input noise temperatures, 

T and T , are derived below, 
ei er 

The objective is to determine the available noise -power output 

from the network, N , in figure A-l, in terms of the various system 

ao 

parameters. 

The (fictitious) noise power N ., is contributed by the network and 

is referred to the network input terminals. This noise power is applied 

through the mismatch plane, M. , [ Rothe and Dalke, 1956; Haus and 

Adler, 1958; IRE Subcommittee 7.9 on Noise, 19&0; Bell, 1951; 

Becking, et al. , 1955] ; this condition is shown diagramatically in 

figure A-l. This concept of network noise power or current flowing 

through the generator impedance and hence, through the mismatch 

plane at the network input, is derived from the form of the equivalent 

"noise-source circuit" which is used in the above references. 

The level of the noise power, N ., depends upon the network noise 

ei 

figure, F [Friis, 1944] , and is given by 

N . = (F - l)kT B (A. 8) 

ei n on 

The available noise power from the source or generator is 

N =kT B (A. 9) 

ag g g 

-23 

where, k = Boltzman's constant = 1.3804 x 10 

T = Generator output-impedance (noise-power) temperature 

B = Generator effective bandwidth 
g 

For the general case, involving impedance mismatching conditions, 

and when B < B , the available noise power at the network output, 
n g 

N , is given by 

ao & y 



-145a - 



NETWORK NOISE POWERS AND NOISE TEMPERATURES 



M 



in 



Tg , B g 



%i 



N a g=kTg Bg 



•in 



'din 



LINEAR. PASSIVE 
LOSSY BILATERIAL 
TWOPORT NETWORK 



Bn. F n » L n i T n 



Mjn ^ogB n /B g tN ei )' 
L dn 



'ei < 



-(F n -l)kT B n 



M; n ( T q -Hej)kB n 
.= (L n -l)kT n B n (Mj n =l) 



w = k T e j B n 



* T eoBnV 



>N, 



Mij = M ir 
y 



•on 



Tei=(Fn-OT 

= (L n -l)T n (Mj n = |) 



( T g+ T ei )M. n 
'eo - i . 



•dn 



LOAD 
OR 
Zj» RECEIVER 

(=Z ir ) 



Ndir 



-N 



'dir 



J=(F r -l) k T Bjf 
er JjkTer B if 

(N ao Bjf/Bn+N er )Mj r 
MT eo +T er )B if M ir 



T er =(F r -l) T 



NOTE, B g >B n >B if 



B x = EFFECTIVE PASSBAND, c/s 



i - N din 



ao 



L dn = L n (FOR Mip = I) 
F = NOISE FACTOR 

k = BOLTZ MAN'S CONSTANT 

L n = TRANSMISSION LOSS FACTOR, AS A RATIO 

N a = AVAILABLE NOISE POWER 

N d = DELIVERED NOISE POWER 

T ei T er = EFFECTIVE INPUT NOISE TEMPERATURE , DEGREES KELVIN 

T g J eo = EFFECTIVE OUTPUT NOISE TEMPERATURE , DEGREES KELVIN 

T n = THERMAL TEMPERATURE, DEGREES KELVIN 

T = STANDARD REFERENCE TEMPERATURE 

Z in = INPUT IMPEDANCE AT NETWORK INPUT TERMINALS 

Z on s OUTPUT IMPEDANCE AT NETWORK OUTPUT TERMINALS 

M; = IMPEDANCE MISMATCH FACTOR 



Fig.A-l 



-146- 

N = -r^ (A. 9a) 

a ° L dn 



( (N B /B ) + N . ^ 
V ag n' g ex J 



L 

an 



(A. 9b) 



(N B /B ) + (F - l)kT B ] 

ag n g n o n/ 



L 

an 



(A. 9c) 



For the case of a transmission line or lossy passive twoport 
network, impedance -matched at its input and output. [ Siegman, 1961a, 
p. 86] , see also [Strum, 19^6, 19^8; Grimm, 1959, I960] ; we have 



where, 



N . = (L - l)kT B (A. 10) 

ei n n n 



T = Thermal temperature of the lossy elements of the 
network. 

L = Dissipative transmission loss factor, defined above, 
n 



From (A. 8) and (A. 10) it is evident that 



F = 1 + (L - 1)(T /T ) (A. 10a) 

n n n o 



Effects of Impedance Mismatching 
In the Receiving System 



Non-reflecting impedance -matched conditions are necessary 
throughout a receiving-antenna system, in order to avoid multiple 
reflections and consequent delayed or ghost signals at the receiver 



-14 7- 
input terminals. However, an impedance mismatch at the receiver 

input is sometimes beneficial, provided that proper impedance mis- 
matching at this point results in an improvement of the receiver noise 
figure. When the receiver -input impedance is not matched to the out- 
put impedance of a (matched-impedance) antenna system, the power 
which is reflected from the receiver-input terminals will be transmit- 
ted back through the antenna system where it will be absorbed in the 
transmission line and also re-radiated from the antenna, and multiple 
reflections will not exist in the receiving system. Consequently, if 
impedance-mismatch conditions exist only at the receiver input, inter- 
fering or ghost signals will not be developed at the receiver-input 
terminals. 

From the above discussion it is evident that impedance mismatch 
conditions, at more than one point in the receiving-antenna system, 
cannot be tolerated in a multichannel or broadband system. Hence, in 
the following analysis it will be assumed that non-reflecting impedance- 
matched conditions exist throughout the receiving-antenna system, except 
that impedance -mismatch conditions may (or may not) exist at the 
receiver input. Under these conditions, the "load" in figure A-l is 
replaced by the radio receiver, and the impedance mismatch factor, 
M. (= M ), will be equal to or less than unity. 

Furthermore, in practically all radio receiving systems, the char- 
acteristic impedance of each network unit is essentially "real", hence 
the following conditions apply: 

X /R < < 1 (A. 11) 

n n 

From (A. 11), (A.6d), and the fact that the system is always 
matched at every point looking toward the antenna 

L =L (A.12) 

an n 



-148- 
From the previous subsection and for the conditions given by 
(A. 11), and (A. 12), and also referring to figure A-l, we have 



((N B /B ) + N . ) 
V ag v! g eij 



N = N " 5 " 5 ^ (A. 14) 

ao L 

n 



or (see (A. 10) ), 



N B /B + (L - l)kT B 



ag n g n n n, 

N = A - s-r: *- (A. 15) 

ao L, 



n 



Equation (A. 15) may be used to determine the available noise - 
power output from a system of linear lossy twoport networks in tan- 
dem, under conditions of impedance matching at the input terminals 
of the networks, looking toward the antenna. 

Noise Levels in the Receiving-Antenna System Networks 

Referring now to figure 4-1, the "loss-free" antenna unit by defi- 
nition does not contribute any noise; however, this unit receives and 

has available at its output the noise power N A = N + N . + N from 

^ ^ A at ai a 

the following sources: 

(a) radio-transmitter output amplitude noise, N , associated 
with the received radio-frequency carrier signal; 



-149- 

(b) interfering radio-frequency signals, N .; 

3-1 

(c) the antenna- received environmental noise power plus atmos- 

pheric and galactic noise, all of which add up to give N . 

el 

The effects of the received noise power, N , are discussed in 

at 

Sections 4.2 and 4.3; where it is shown that proper allowances may be 
made for the effects of this noise power by using the results of equip- 
ment back-to-back tests to modify the (ideal) system-design curves. 
Therefore, N , need not be included at this point in the analysis of 
noise powers in the receiving system. 

Also, the case of interference (noise) signals is not considered 
in this report. The effects of interfering- signals are dependent upon 
a variety of factors, including the "capture -effect" in FM systems; 
with the result that the interfering- signal noise power, N ., is not 

3,1 

simply additive. Hence, for our purpose the available noise power at 
the output terminals of the loss -free antenna is 

N A = kT B (A. 16) 

A a a v ' 

where B is the effective bandwidth of the receiving antenna, and T 
a & a 

is the antenna received-noise temperature. 

Each of the passive lossy units, in the receiving-antenna system, 
contributes noise power, and a portion of each of these noise powers 
is subsequently available at the duplexer output (receiver input) term- 
inals. The total available noise power at the duplexer output, due to 
the noise contributed by each lossy antenna unit plus the antenna-temp- 
erature noise, is designated here as N and is assumed to be uniformly 

& au ' 

distributed over the effective bandwidth of the duplexer, B n . 

du 

Interfering- signal noise power is also added to the receiving system 
from the local transmitter through the duplexer. The level of this 
noise power at the duplexer output (receiver input) terminals, is 



-150- 
dependent upon the power output from the local transmitter, P , and 
the "isolation factor" of the duplexer, Lp. The available local-trans- 
mitter noise power level at the duplexer output (receiver input) term- 
inals is _ 

p <t 

N it =— (A. 17) 

where, P = Power output of the local transmitter, in watts. 

L = Transfer isolation factor of the duplexer. This 

factor is defined as the ratio of P to that portion 

of P. which is available at the duplexer output 

(receiver input) terminals, and is within the receiver 

RF spectrum effective bandwidth B -( = B. r ). 

rf if 

The position of the main portion of the RF spectrum of P„ must 

r r ^ it 

be well outside of the RF spectrum included by B and the receiver 

"image-frequency" spectrum. Note that in the above definitions the 

available noise power, N , is distributed uniformly over the effective 

au 

bandwidth of the duplexer, B, , while the available noise power N. is 

r du ^ it 

(assumed) to be distributed uniformly over the effective bandwidth of 

the receiver, B . or B... The selectivity characteristics of the duplex- 
rf if 

er and the receiver-input circuits are relied upon to exclude the local- 
transmitter power (outside the bandwidth B .) from the receiver IF 

rf 

circuits. 

The sum of the noise powers in the receiving-antenna system at 

the duplexer output terminals is N = N + N . This noise power, 

N > is available at the duplexer output, that is, at the receiver input 

terminals. 

Because of the manner in which L. has been defined, N, is the 

D it 

noise power from the local transmitter as measured by the receiver 
having an effective bandwidth B . The value of N. will depend upon 



-151- 
the position of the effective bandwidth of the power-measuring device, 
(in this case the receiver) relative to the position and the shape of 
the power spectrum of P. . 

Effective bandwidths of the cascaded units in the receiving-antenna 
system are assumed to be greater than the effective bandwidth of the 
receiver, B . Hence, as noted previously, the noise power "accept- 
ed" at the receiver-input terminals will be proportional to the receiver 

effective bandwidth, B.,., and to the noise-power densities, N /B , , 

if au du 

and N. /B . Furthermore, if impedance-mismatch techniques are 

employed at the receiver -input terminals, in order to minimize the 

receiver noise figure F , then the portion of N which is delivered 

to and absorbed at the receiver input terminals is dependent upon the 

value of the impedance-mismatch factor, M. . Therefore, the total 

lr 

receiving-antenna system noise power delivered to the receiver- 
input terminals, N , and subsequently processed in the receiver and 
transferred to the receiver IF circuits, is 

/ N au N it\ 

N^ = M. ( -^ + -ii )B. £ (A. 18) 

Td irV B n B.,. i if 

v du if / 



In (A. 18), M. is the impedance mismatch factor at the receiver 



input . 



In terms of available power at the duplexer output, M is 

lr 

4 R R. 
(M. ) = ^-i r , (A. 19) 

irav |z, +z. | 2 

du lr 



In terms of non- reflecting matched-impedance power at the duplexer 
output, M. is 

du du ir 



(M. ) = ? (A. 20) 

ir mat i „ . „ i £- n 



Z, +Z. ! R, 
du ir du 



-152- 

where, 

R, = Resistive component of the output impedance of the 

du j i 

duplexer. 

Z = Output impedance of the duplexer. In general, this 
is the output impedance of the network which pre- 
cedes the receiver. 

Z = Input impedance of the receiver, 
ir 



In order to carry out the above procedures, and obtain an estimate 
of N , the noise power output of a linear passive lossy bilateral net- 
work, N , is determined in terms of the noise power input to the unit, 
ao 

N , the thermal temperature, T , transmission loss factor, L , and 
ag n a 

the noise figure F of the unit; see figure A-l and (A.1'5). 

Referring to figure 4- 1, the output noise powers from each of the 
tandem units may be obtained by substituting the proper factors in 
(A. 15). This procedure will be used in a later section of this Appendix 
to calculate the value for N , the total noise power at the duplexer 
output terminals, preceding the receiver input terminals. 



Receiver Noise 

The noise power contributed by the receiver [ Bell, R. L. , 195l] 
[Becking, et al. , 1955] , [ Rothe and Dalke, 1956] , [ IRE Subcommittee 
7.9 on Noise, 19&0] and delivered to the receiver-input terminals is 
very closely approximated by, 

M. N = M. (F - l)kT B. r (A. 21) 

ir er ir r o if 



where, 



M. = Impedance mismatch factor at receiver-input term- 
inals, as given by (A. 16) 

F = Noise figure of the receiver [ IRE Standards, Proc. 
IRE, Jan., 19&0] , as a ratio. 



-153- 

The receiver IF effective bandwidth, B.., is defined by: 

if 



> if = "^- Cg (f)df (A. 22) 

r 



where, 



G = Effective receiver power gain, as a ratio; this 

factor is defined as the ratio of the signal power 
delivered to the receiver input terminals to the 
available output signal power in the receiver pre- 
detection or IF circuit at an arbitrarily specified 
frequency. This frequency may correspond to 
that of maximum gain. 

G(f) = Receiver power gain as a function of the frequency, f. 

Since our results apply to radio communication type of receivers, 
B , is assumed to be the effective single-response IF channel band- 
width. 

The receiver noise figure, F , as used above must be measured 

r 

using a "source" or generator output impedance which is equal to 

Z. . In practice, the ratios X n /R n and X. /R. are each very 
du du du lr lr 

much less than unity, and hence conjugate -impedance matching and 
non- reflecting impedance matching are equivalent. Furthermore, 
the signal generator or source impedance, Z , may be varied in 
order to obtain the optimum or minimum value for the receiver noise 
figure, F [Bell, 1951] , [Becking, et al. 1955] , [ Rothe and Dalke, 
1956] , and [ IRE Subcommittee 7.9 on Noise, Jan. , 19&0] ; the par- 
ticular value of Z which gives F would establish the correct value 
for Z required to insure optimum impedance-mismatch operating 
conditions at the receiver input. 



-154- 

Pre-Detection Noise Power, N.- 

if 

The receiver-contributed noise power, M. N , is added to the 

lr er 

delivered noise power, N (see (A. 18)), from the receiving system 

which precedes the receiver, and this total noise power level is 

multiplied by the receiver power gain.G , to obtain the predetection 

r 

noise power, N , that is, 



N if= < N Td +M ir N er )G r < A ' 23 > 

The pre-detection noise power, N. f , obtained from (A. 23) applies 
only to a linear receiver having sufficient selectivity, in the input 
section, to eliminate "image-frequency" or "spurious-response" 
(signal and noise) power in the receiver IF circuits. The total IF 
noise power in multiple -response receivers depends upon the number 
of channels accepted by the receiver; this total IF noise power is 
equal to the sum of the noise power contributed by all of the channels. 
In radio communication systems only the "principal-response" chan- 
nel is used. In receivers employing either or both age and limiting, 
the receiver power gain, G , will be a function of the power level of 
the receiver-input RF signal, P . 

The above ideas are now used as a guide in determining the funct- 
ional relationship between the pre-detection noise power level, N , 
and the receiving-system parameters. The predetection load-reflect- 
ed noise power [ Siegman, 19&lb] referred to in Section 4.2 is assumed 
to be negligible, and is not included in the following analysis. How- 
ever, this factor would be important for very-low-noise bilateral 
receivers with an impedance-mismatch load connected to the receiver 
IF output terminals. Referring to figure 4-1, and substituting (A. 18) 
and (A. 21) in (A. 23) we obtain the following relationship for N , the 
available pre-detection noise power, 



N e = G 
if r 



-155- 

N i ; . 

(A. 24) 



it\ 



M + — -]B , + M. (F - 1) kT B., 

lA B du B if^ lf ir r ° lf 



Each of the two terms inside the overall brackets on the right- 
hand side of (A. 24) represents components of the total noise power 
delivered to the receiver-input terminals. The noise power contribu- 
ted by the receiver, and delivered to the receiver input [ IRE Subcom- 
mittee 7.9 on Noise, i960] is given by , M. (F - l)kT B. r ; 

ir r o if 

see (A. 21 ). The noise power, N' = (N /B , + N„ /B.,)B.-, is com- 
1 >. ^ T au' du it' if if 

posed of the noise power received by the antenna, N , plus the noise 
power generated within the lossy passive networks in the receiving- 
antenna system; including the noise power N. , which is contributed 
by the local transmitter; N' is available within the RF spectrum band- 
width B „ (or B..) at the receiver input terminals. This noise power, 
rf if 

N' , is available at the output of the network which precedes the 

receiver, within the RF spectrum bandwidth B ,(or B.J accepted by 

rf if 

the receiver, and is given by 



N' = 
T 



N N, -1 

au it 

^ B du " B i£ - 



B., (A. 25) 

II 



Note that N' - N B /B only when the various noise powers in 
the receiving -antenna system are uniformly distributed over the ef- 
fective bandwidth of the duplexer, B, . 

du 

The factors within the brackets of (A. 25) may now be determined 
in terms of the receiving antenna- system parameters, using the ideas 
outlined above to estimate the total noise power at the output of the 
tandem units in the antenna system, as follows: 



-156- 
From (A. 15), (A. 17), and figure 4-1 we obtain 



N 



N 



au 



it 



B 



du 



N B n /B (L -l)kT B, 
L du t du du du 



du 



du 



it 



B 



du 



if D 



N 



B /B +(L -l)kT B B A (L -l)kT, 

c t ac t t t \ du du du du 



B 



L L n 
t du 



J-\ + 



du 



B 



du 



it 



B if L D 



(N A B /B ) + (L -l)kT B B (L - l)kT B 

A ac' a c c ac t t t t 



c t du 



ac 



L L 
t du 



du 



(L -l)kT B 

du du du 



it 



B 



du 



if D 



(A. 26) 



Since N. = kT B , see (A. 16), (A. 26) may be written as 
A a a 



N N 

au x-t 



du if 



T + (L - 1)T (L - 1)T (L, - 1)T, 

a c c _t t_ du du 



L L L 

c t du 



t du 



du 



it 



if D 



(A. 27) 



where, 



L , L , L 
c t du 



Transmission (dissipative) loss factors for the 

antenna circuit, transmission line, and the 

duplexer network, respectively; as a ratio^ see p. 143 



^Antenna-earth "proximity effects" on the antenna circuit losses 
[Wait, J. R. , 1959; Vogler, L. E. , 19&2] , are included in the 



transmission loss factor, L . 

c 



-157- 
The noise power available to the receiver within the receiver 

IF bandwidth, B. r , is given above by (A. 25) as: 

if 



N' = 
T 



r N N. 

au , it 



- B du 



if J 



B. 



(A. 25) 



Re-arranging terms in (A. 27) and substituting in (A. 25), we 
obtain 



T 



r T + (L - 1 ) T + L (L-l)T + L L (L, -1)T, P^ 

r a c c c x t ' t c t x du ' du it 



c t du 



B if L D ^ 



kB 



Equation (A. 28) may be written as 



(A. 28) 



N' = kT B. r 
T A if 



(A. 29) 



From (A. 28) and (A. 29) we obtain: 



r T + (L - 1)T + L (L - 1)T + L L (L J - 1)T J 
a c c c t t c t du du 

c t du 



it 



kB i£ L D 



(A. 30) 
The factor, T , is the "effective receiving antenna- system (out- 
put) noise-power temperature", and is completely defined by (A. 29). 
Because of its fundamental nature and simplicity of application, the 
concept of effective noise temperature is in general use in the fields 
of radio communication and radio astronomy, and is referred to 
throughout the literature in these fields. 

From (A. 29) and (A. 30) it is seen that T is a measure of the 
total noise power received by the antenna, plus the noise power origi- 



nating within the receiving antenna system. The noise power, NL, is 



-158- 

" available" at the duplexer output terminals. The portion of N' that 

lies within the bandwidth B is processed by the receiver and is 

available at the "pre-detection" point within the receiver. 

The factors * L , L , L, , and L^ in (A. 30) have been defined 
c t du D 

previously. T will be in degrees Kelvin if T , T , and T are in 
J\ c t du 

degrees Kelvin, and also if P. is in watts and B. r is in c/s of band- 
it if 

width. 

The effective antenna-system output noise temperature, T , as 
determined from (A. 30) is seen to depend upon: (1) received noise- 
power temperature of the antenna, T , (2) thermal temperatures of 
the units in the receiving-antenna system, T , T , and T , (3) 

receiving antenna- system transmission loss factors, L , L , and L, , 

c t du 

and (4) other factors such as, the receiver IF bandwidth B , the 

power output of the local transmitter, P. , and the isolation factor of 

the duplexer, L„. 

It should be noted that (A. 30) determines the value of T . for un- 

A — 

restricted values of the thermal temperatures of the antenna- system 

networks, T , T , and T . Hence, this equation is applicable to the 
c t du 

case where the networks in the receiving-antenna system are at a low 
thermal temperature; such as might be achieved with appropriate re- 
frigeration, or perhaps in a "space" receiving system. 

It should also be noted that (A. 30) is restricted to the case where 
non- reflecting impedance matched conditions exist throughout the 
receiving-antenna system, when looking towards the antenna. The 
receiver input, may or may not be matched; (A. 30) is further re- 
stricted to the case where, X /R < < 1. 

n n 

♦ See discussion of L on p. 142. 

n 



-159- 

The effect of the loss factors, L and L. , on the antenna- system 

c t ... 

v 

temperature, T , may be seen by applying (A. 30) to a typical example. 

The receiving antenna system is assumed to consist of only an antenna 

and a transmission line, with the following paramaters: 

T = 300 degrees Kelvin 
T a = T = 290 degrees Kelvin 
L c =1.12 (= 0.5 db) 
L C = 1.58 (= 2. db) 

From the above specified parameters and (A. 30), we obtain 

300 + (1.12 - 1) 2 90 + 1.12 (1.58 - 1) 2 90 
A " 1.12 x 1.58 



= 170 +19.6 + 106 = 295.6 deg. Kelvin 

The noise-temperature contribution from each unit in the antenna 
system is evident from these calculations. The calculated noise temp- 
erature, T (= 295.6 K) is seen to be only slightly different from the 
assumed antenna temperature, T (= 300 K); hence, the loss factors, 
L and L. , have the dual role of effectively reducing the antenna temp- 
erature, T , and also contributing to the resultant noise temperature, 

cL 

T , at the receiver input terminals. 

However, the receiver-input signal, P , will also be reduced by the 

r 

product of the loss factors, L L , (= l/l.l2 x 1 58 = 0.56), and the net 
result will be a degradation of the predetection signal-to-noise ratio, 

W 

Equation (A. 30) may be simplified if the thermal temperatures of 
the receiving-antenna passive-network units are each at the same 
(standard) temperature, T , and if impedance -matching conditions look- 
ing toward the antenna, exist throughout the receiving-antenna system; 

that is: T =T =T, =T (A. 31) 

c t du o 

* In this example a duplexer was not used. 



-160- 

Substituting conditions (A. 31) in (A. 30) and combining terms, we 
obtain: 

T - T P 

T . = T~^ — T 2 — +T + , „ T (A. 32) 

c t du if D 

If (a ) the thermal temperature of each of the receiving-antenna 
elements is T , (b) X /R < < 1, and (c) non- reflecting impedance- 
matching conditions exist throughout the receiving-antenna system 
looking toward the antenna, the effective antenna- system output tem- 
perature, T , maybe calculated from (A. 32). 

Equation (A. 2 3) may now be written as, 

N. r = G M. [T + (F - 1)T ] kB m (A. 33) 

if r lr A r o if 



where F is the receiver noise figure, (see p. 153), and M. is de- 
r i r 

fined by (A. 16), for "available" noise power at the duplexer output. 

The factor, (F - 1)T , in (A. 33) is the effective receiver (input) 
r o 

noise temperature, T , [ IRE Comm. on Noise, 19&0] . Hence (A. 33) 

er 

may be written as, 

N. r = G M. [T A +T ] kB.. (A. 34) 

if r ir A er if 

Equation (A. 34) is used throughout this paper to estimate the 
receiver IF noise power level, N . 

The performance of a radio receiver depends upon the receiver 
IF total signal power-to-total noise power ratio, P /N . An inspect- 
ion of (A. 34) shows that the receiver IF noise power level, N , depends 
upon both the receiving antenna-system noise power level, kT ^.-, 

and the receiver-contributed noise power level, kT B... 

c er if 



-161- 

The noise power, kT B.., may be assumed to be thermal noise 

er if 

with a steady average power level. In the frequency range above 
(approximately) 100 Mc, the received noise power which is associated 
with the antenna noise temperature, T , may also be assumed to be 
thermal noise. Therefore, for radio communication systems oper- 
ating with RF signals above 100 Mc, the receiver IF noise, N , may 
be assumed to be "white", and its average total power level may be 
estimated from (A. 34). In the frequency range well below 100 Mc, 
the noise signal, kT B , may have the characteristics of atmos- 

3- IX 

pheric noise. Under these conditions, N , may be estimated from 
(A. 34) provided that kT B is assumed to be an "average" value; 
obtained by averaging over a period of time which is long compared 
with the average period of time between noise peaks. 



-163- 

APPENDIX B 

Derivation of the Message-Channel Signal-Power Loading 
Ratio P /p , in the Modulating Baseband Signal 



Assuming that the message -channel signal power S , at the radio 
receiver output, is proportional to the modulating signal-power load- 
ing ratio, p /P , we have: 
6 *m m 

P m 
S = K, x G x-^=- (B. 1) 

oc 1 r P x 

m 



where, 



S = a particular message-channel average signal power 
level, at the radio receiver output, or decoder input. 

K - constant of proportionality, dependent upon the radio 
system parameters. 

G = gain of the linear section of the radio receiver. 

r ° 

p = average power level of a particular message signal 



m 



m 



in the composite modulating baseband signal. 

average of the total power level of the composite modu- 
lating baseband signal. 



Note that for SSBSC time -division message -signal multiplexing 
systems which are being considered in this paper, p will be confined 
within a definite bandwidth in the baseband- signal spectrum and may 
be measured directly in the baseband. For other types of multiplexing 
systems (see Subsection 4. 4) it may not be possible to measure or 
adjust p directly in the modulating baseband; hence, modifying factors 
might be required in (B. 1). 

The message-channel average noise-power level at the radio re- 
ceiver output is, 

N = K x G x B (B.2) 

oc Z r c 



-164- 
where, K = constant of proportionality, dependent upon the radio- 

system parameters. 



B = bandwidth of a particular message-signal channel, at 
the receiver output, in c/s. This bandwidth is assur 
to be equal to the spectrum of the message signal. 



Since the radio receiver gain, G , is dependent upon the receiver- 
input radio-frequency signal power level, it is inconvenient to use 
either (B. 1 ) or (B. Z) separately. Hence, by combining (B. 1 ) and (B. Z) 
we obtain the message-channel signal-to-noise ratio at the radio re- 
ceiver output, S /N , and also eliminate the variable radio- receiver 
oc oc 

gain factor, G . Combining (B. 1) and (B.Z), we obtain, 
r 

oc m 

The "constant" K in (B. 3) is a function of the radio system param- 
eters; its functional form and its value depend upon the type of radio 
system--see Sections 4.5 and 4.6. 

From (B. 3) we obtain, 

p B (S /N ). 

ml cl oc oc 1 



p " B (S /N ). (B * 4) 

mZ cZ oc oc Z 



or, in general; 



p . B (S /N ). 

ml cl oc oc 1 

~ " B . (S /N T 

mi ci oc oc l 



(B.5) 



and, 



p . B .(S /N ). 
rni ci oc oc l ._ , . 

r- = B..(S. ,/N ) (B - 6) 



mn en oc oc n 



-165- 
From (B. 4) to (B. 6) we obtain: 



B AS /N ). 
cl oc oc 1 

P ml" P mnB (S 7n T (B.7) 



en oc oc n 



or, in general: 



B _(S /N ). 
cZ oc o c 2 . . 

m2 = P mn B (S /N ) {B ' 8) 

en oc oc n 



B .(S /N ). 

Cl OC OC 1 .„ .. 

"mi^mn B (S / N ) (B - 9) 

cn oc oc n 



[B (S /N )] a 
c v oc' oc /J A 

P mA = P mn B (S /n ) (B ' 10) 

cn oc oc n 



The subscript A denotes a "class" of message signals, each class 
ires the same value of the product [ 
receiver output. Continuing, we obtain: 



requires the same value of the product [ B (S /N )] , at the radio 

c oc oc 



[ B (S /N >] „ 

c x o& oc' J B mm 

mB = P mn B (S /n ) (B - H) 

cn oc oc n 



[ B (S /N )] ^ T 

c v oc / oc /J N ._. 

P mN = P mn B (S /N ) (B ' 12) 

cn oc oc n 



The total power level, P , in the modulating baseband signal is, 



m 



P = M A p . + M^ p „ + ...+ M nvT p (B.13) 

m A *mA B y mB N ^mn v 



-166- 

where, M = number of simultaneously- active message signals 

each of which requires the same value for the product 

[B (S /N )] 
c oc oc A 

M„ = Ditto, for the product [ B (S /N )] . 
B ^ c oc' oc ,J B 

M„ = Ditto, for the product [ B (S /N )] . 
N c oc oc N 

Combining (B. 10) to (B. 12) and (B. 13) we obtain, 



m 



P / /S 
mn / r f oc 

B (S /N ) ( M A [B c( 5T 



cn oc oc n 



oc 



K +m Jb/-^\ + ... 



B L c^N 



oc 



x oc' x oc 



(B.14) 



Combining (B. 6) and (B. 14) we obtain: 

S % .S 



m 



mi 



x OC' x OC' x oc / 



N 



B .(S /N T 

Cl OC OC 1 



(B.15) 



Equation (B. 15) is reproduced as (4. 9) in Section 4. 3. 2, 



-167- 
APPENDIX C 

Derivation of FM Receiver Performance Characteristics 



For the condition where the total signal power-to-total noise 
power ratio, P. f /N. f , at the IF output, that is, at the input to the lim- 
iter of a "conventional" type FM radio receiver, exceeds (approximat- 
ely) 10 db, it is possible to calculate the relationship between the signal- 
to-noise ratio, S /N , at the receiver output, and the ratio of the radio- 
o o 

frequency signal power level, P , to the total noise power at the receiv- 
er input; k (T + T )B . These calculated results apply only to the 
upper linear portion of the FM receiver characteristic curve, above 
the threshold level. The following deriviations indicate the method 
used to calculate the upper linear portion of the FM radio receiver 
characteristic curve, and the parameters to be considered. 

The average total noise power in a one-ohm circuit at the output 
of an FM discriminator [Black, 19^3] through a low pass filter having 
a cutoff frequency of f c/s is: 



4 * d r 3 l f 

N = — - — x — — x f n x — — 

o 3 P.. k B. r 

if if 



(C.l) 



where, 

D n = demodulator conversion factor, 
d 

P = total average IF signal power at the receiver limiter 

input, in watts. 

N = total average IF noise power at the receiver limiter 
input, in watts. 

B. = effective bandwidth of the receiver IF circuits, 
if 

Equation (C.l) indicates that the discriminator-output noise volt- 
age spectrum is triangular, increasing with frequency up to the cutoff 
frequency f . This equation also shows that the total discriminator- 



\ 



-168- 

output noise power is inversely proportional to the IF signal power P.,» 

and directly proportional to the noise power density N./B., in the IF 
passband. 

From (C. 1) the total average noise power out of an FM discrimi- 
nator in a message-channel passband from f to f is: 

6 * a b 

N oc = — X P- X(f b ' V X IT (C - 2 » 

if if 

Equation (C.2) gives the difference in discriminator -output noise 
power levels between the triangular-shaped output-noise voltage 
spectra having cutoff frequencies f and £ . 

The average signal power in a one-ohm load at an FM discrimin- 
ator output, associated with a sinusoidal modulating signal of frequency 

f is [Black, 1953] : 
c 

S = 2 it 2 D^ AF 2 (C.3) 

oc d c 



where, AF = peak frequency deviation of the radio frequency car- 
rier, in cycles per second, due to the sinusoidal modu- 
lating signal of frequency f . 



Combining (C.2) and (C.3) we obtain, 

S _ AF 2 B. f P.. 

— = - x , C * x -^ (C.4) 

N 2 3 3 w 

oc (f - f ) if 
b a 

The value of AF is determined by the power level, p , of the 
c m 

modulating sinusoidal signal at the radio-transmitter frequency-modu- 
lator input and the modulator characteristics. This modulating power 
level may be adjusted, relative to the total baseband- signal power level 
so that: 



\ 



-169- 



Vz 



AF = (p /P )' AF (C.5) 

c mm 



where, 

p = average power level of a particular sinusoidal modu- 
lating signal in the baseband; the frequency of the 
modulating signal being f c/s. 

P = total average modulating- signal power level in the 
baseband signal. 

AF = peak radio-frequency carrier deviation in c/s for a 

single sinusoidal modulating baseband signal, having 

an average power level equal to the total average 

power level of the total baseband signal, P 

m 

The message-channel signal-power loading ratio, p m /P » is 

determined by methods outlined in Section 4.4.2 and Appendix B. 

We are concerned with a comparatively narrow portion of the 

baseband spectrum and hence f ~f, . For this condition it can be 

a b 

shown that, 



The portion of the baseband- signal spectrum which is given by 

(f - f ), is related to the message -signal bandwidth B , as follows, 
Da c 



f - f = K B (C.6a) 

b a M c 



where, 



K = factor which depends upon the type of multiplex 
M 

system. 

B = message- signal channel bandwidth, in c/s. 



-170- 
For SSBSC frequency-division multiplex, being considered in this 



paper, K = 1 therefore, 
M 



f K" f = B . (C.6b) 



b a c 



The term (f + f,)/^ in (C.6) is the mid-frequency, i t between 



c 



f and f_ , hence 
a b 



= f (C.6c) 



2 c 

Substituting (C.6b) and (C.6c) in (C.6) we obtain, 



where, 



f 3 - f 3 = 3 B f 2 (C.6d) 

b a c c 



f = Center frequency of the message channel in the 
baseband signal spectrum, in c/s. 



The radio-frequency spectrum bandwidth (B .) , associated with 

rf c 

particular values of A F /f and f , is given by: 

(B ) =f x <^(AF /f ) (C.7) 

rf c c c c 



However, the value of B ., is determined under conditions where 

rf 

all of the modulating- signal power is in a single sinusoidal modulating 

signal having a frequency = f ; tha 

AF = AF; hence, B . is given by 
c rf 



signal having a frequency = f ; that is, p /P = 1, f = f , and 

m m m c m 



B r =f x <)> (AF/f ) (C.8) 

rf m m 



-171- 

where, 

B . - Radio-frequency spectrum bandwidth which includes 

3TX 

all components or sidebands having amplitudes equal 
to or greater than one percent of the unmodulated 
carrier amplitude. 

cf) (A F/f ) = function of A F/f . This function gives the number of 
significant sidebands in the radio-frequency spectrum, 
assuming that the modulating signal is a single sinus- 
oidal voltage wave having a frequency f , in c/s. 

The function <j> (A F/f ) in (C. 8) has been calculated [Hund, 1942] 

m 

and is shown in figure 3-1. 

Assuming that the FM receiver IF bandwidth, B , is adjusted to 
be only wide enough to pass the significant sideband components in the 
radio-frequency spectrum, we have, 



B = B (C.9) 

rf if 



Therefore, 



B. r = f x <j> (AF/f ) (CIO) 

if m m 



Substituting (C.5), (C.6d) and (C.10) in (C.4) and re-arranging 
terms we obtain: 



S oc B c P m 1 ,/AF\ /AF \ 2 P if . 

X J— X = - X <j)(- X ( -r— X — — (C.ll) 



N f p 2 T Vf J V f J N 

oc m m x m /v c / 

The term P /N in (C.ll) is the required predetection total signal 

power-to-total noise power ratio associated with a specified value of 

S /N at the receiver output, and for particular values of the param- 
oc oc 

eters: f , B , p /P , cj>(AF/f ) and (A F/f ). However, in radio 
m c m m m c 



-172- 

communi cation system design work and system performance testing it is 

usually more convenient to deal with the total radio-frequency signal 
power and the total noise power at the receiver input. Hence, the form 
of (C. 11) was modified as follows: 



P. r = G M. P (C.12) 

if r lr r 



From (A. 30) in Appendix A and (C. 10) above, we have, 

N. r = G M. [T.+T ] kf x $(^-\ (C.13) 

if r lr A er m If 7 



m 



therefore, 



where, 



P., P 

if r 

NT " [T A + T ]~kf x <(> (A F/f ) (C14) 

if A er m m 



G = power gain of the cascaded RF and IF linear sections 
of the FM receiver. 

P = total average radio-frequency signal power at the 
receiver input, in watts. 

-23 

k = Boltzman's constant = 1.3804 x 10 joules per degree 

Kelvin. 

f = highest frequency in the modulating baseband signal. 

T = receiving antenna- system output noise temperature, 
degrees Kelvin; see Appendix A. 

T = radio receiver-input effective noise temperature [ Proc. 
IRE Comm. on Noise, I960] , degrees Kelvin. 



er 



Substituting (C.14) in (C.ll) we obtain, 



s B P , A ^ x 2 P 
oc cm/ AF \ r . <-. 

X X = ( — \ X — ; t— — — rrz (C.13) 



N f p \ £ J 2k(T A +T)f 

oc m rn v c ' A er m 



-173- 
Note that the left-hand side of (C.15) coincides with that of (C.ll); 

this similarity was preserved in order to compare the predetection 

total signal-to-noise ratio, P /N , with the RF signal-to-relative 

noise power ratio, P /k(T . + T )f , at the receiver-input terminals. 

r A er m 

It is evident from (C.15) that the receiver-output signal-to-noise 

ratio, S /N , above the threshold region, is inversely proportional 
oc oc 

to the square of the modulating frequency f , for a given set of system- 
parameter values. This situation does not apply to the receiver per- 
formance in the threshold region; in this region the receiver -output 

sisnal-to-noise ratio, S /N , approaches the same value for all 
5 oc oc *^ 

frequencies in the modulating baseband signal. Hence f in (C. 15) was 

made equal to f , the maximum frequency in the modulating signal, 

in order to obtain a design equation which is conservative above the 

threshold region. Equation (C. 15) is reproduced as (4.19) in Section 

4.6.1.1, withf = f . 
c m 



-174- 

APPENDIX D 

Method of Estimating the Performance Characteristics of Radio 
Receivers and Message-Signal Decoding Units, 
for a Time-Varying Signal. 

Estimated equipment performance, for a time-varying signal, is 
outlined by a method of graphically combining the steady- signal per- 
formance characteristic of the equipment with either the distribution 
of the time-varying signal power or the distribution of the time-vary- 
ing signal-to-noise ratio at the equipment input. 

The analysis of radio receiver performance in this Appendix does 
not include system intermodulation noise; see Section 4.3. Hence, the 
estimated performance will apply to the "ideal" receiver character- 
istic, as shown in figure 4-A. Proper modifications of the estimated- 
performance results may be made for the effects of system-intermodu- 
lation noise, using the procedures outlined in Section 4. 3.6. 

D. 1 Estimated Performance Characteristics of Radio Receivers. 

The total post-detection signal-to-noise ratio, S /N , at the out- 
^ B o' o 

put of an "ideal" radio receiver depends upon the total received radio- 
frequency signal power, P , and the total noise power, N' + N , 
available at the receiver input. Where N' is the antenna- system noise 
power which is available at the receiver input terminals, and lies with- 
in the RF signal spectrum bandwidth B A=B. r ). N is the noise con- 
5 ^ rf if er 

tributed by the receiver, referred to the receiver input. See Appendix 
A. N' + N may be estimated by adding the antenna- system noise 
and the receiver noise on a power basis and the resultant (for the case 
of radio systems operating at and above VHF) is assumed to have the 
characteristics of "white noise", with a steady average value. In the 
following derivations the total signal-to-noise ratio, S /N , is convert- 
ed to the message -channel signal-to-noise ratio, S /N (see Section 
6 6 oc oc 

4), in order to conform to symbols used in Sections D. 1 and D. 2. 



-175- 

In general, the statistical characteristics of both P and N must 

r T 

be considered. The characteristics of P to be considered are (1) the 

r 

cumulative (time) distribution and (2) the fade rate. The character- 
istics of N to be considered are (1) the cumulative (time) distribution, 
including average and median power levels, and (2) the spectral density 
in the frequency range of the spectrum included in B f . 

In this work the received RF signal, P , is assumed to be of the 

r 

Rayleigh-fading type. Other types of received signals may be con- 
sidered, provided that their distributions are known and also provided 
that the performance characteristics of the combiner are known for 

these other distributions. The fade rate of P at its median level is 

r 

assumed to be lower than the lowest binary-bit rate in the modulating 
message signal. In other words, it is assumed that there is no apprec- 
iable change in the value of P during the duration of a message -signal 
- x 

binary pulse, or during the period of time required for one cycle of 

the lowest frequency in the modulating (message) signal. 

At this point it should be noted that the characteristics of the 

receiver -output noise, N , for an FM receiver, will depend upon the 

oc 

level of operation of the receiver with reference to the "threshold re- 
gion". Hence, this situation must be considered when determining the 
performance characteristic of FM receivers. 

The method of determining the receiver performance character- 
istic for a time-varying received RF signal, P , is based on the idea 
of combining the receiver-input steady RF signal performance character- 
istic with the cumulative power-level distribution of the received RF 
signal. Following is the detailed procedure for applying this method 
to determine the performance characteristic of an FM receiver for 
a Rayleigh-fading type of received RF signal. 



-176- 
Figure D-la shows the measured steady-signal performance char- 
acteristic for an FM receiver, covering a range of receiver-input 

values of P /f (T . + T ) and receiver output message -channel 
r' m A er' r 6 

signal-to-noise ratios, S /N . Normalized abscissa and ordinate 

oc oc 

scales are used to conform to other work in this report; see Section 

4.6, figure 4-3. The normalizing factors are: f , T A , T , B , and 

m A er c 

P /p . Note that A F/f = 1 for this curve and hence curves derived 
mm m 

from this characteristic, for a time-varying receiver -input relative 

z AF 

signal-to-noise ratio, P /f (T . + T ), apply only to — — = 1. P is 
6 r' m A er ff y y ^ r 

m 

the power level of the total RF signal, available at the receiver-input 

terminals. The total receiver-input noise consists of thermal noise 

from the signal generator and from the receiver; hence T. = T (gen- 

A g 

erator output noise-power temperature) and T = (F - 1) T . N' + N 

^ ^ ^ ' er v r o T er 

may be obtained from: N' + M - k (T A + T )B. r . Note that the 
7 T er A er if 

relative value of N' + N (= k(T A + T )f ) is used in the abscissa 
T er v A er m 

titles in figure D-l. 

Curve A-A in figure D-lb gives the cumulative distribution of the 
Rayleigh-fading received RF signal, P . Any median value may be 
selected for P by proper adjustment of the ordinate scale values. 

The method of obtaining the estimated FM receiver performance 
characteristic for a time-varying received RF signal is as follows: In 
figure D-lb, an abscissa value of percentage of time t is selected, and 

3. 

the corresponding normalized ordinate- scale value P' , is noted. The 
value of P 1 from figure D-lb was used in figure D-la to determine an 
ordinate- scale value on the normalized scale; this normalized signal- 
to-noise ratio is denoted here as (S /N )'. This value of (S /N )' 

oc oc oc oc 

was then plotted versus t as shown in figure D-lc. The above pro- 
cedure was repeated to obtain additional points for the curve in figure 
D-lc. 

The average value of the distribution of the relative signal-to-noise 
ratios was obtained by numerical integration of the curve and is shown 

in figure D-lc as (S /N ) . Note that the integration must be 
& v oc' oc'ave 



-177- 



GRAPHICAL METHOD 
FOR DETERMINING FM RECEIVER PERFORMANCE 



F 






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10 LOGiq P r - 10 LOGio f m - 10 LOGio (T A + \ r ) 

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RAYLEIGH- FADING NON-DIVERSITY 
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(Pr)med 





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lOUOGm P r -lOLOGKjfm -K)LjOG io (T a +Tg r ) 

Fl» D-ld 



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'o 







001 I N 30 50 TO 90 95 99 995 99.9 99S5 99 99 

PERCENT OF TIME ORDINATE IS EXCEEDED 

Flfl. D-lb 



aa i to so so n so 15 nsu susms mm 
PERCENT OF TIME ORDINATE IS EXCEEDED 

Fl»0-lc 



-178- 
carried out using ordinate -scale values as numerical ratios instead of 

decibel units. This value of (S /N ) , together with the value of 

oc oc ave. 

P (med.)from figure D-lb, were plotted as shown in figure D-ld. Add- 
r 

itional points for the FM receiver performance characteristic curve in 
figure D-ld were obtained from similar distributions (curve A-A) in 
figure D-lb but with different P (med. ) values. The resultant FM 
receiver-performance characteristic curve in figure D-ld is in terms 

of the median of the time -varying received RF signal, P , and the 

r 

average of the distribution of the time-varying receiver-output relative 

signal-to-noise ratio, S /N . The shape and the position of the 
B oc oc r ^ 

curve in figure D-ld is related to the shape and the position of the 
curve in figure D-la in terms of their respective abscissa and ordinate 
scales; hence, the accuracy of the estimated receiver-performance 
curve in figure D-ld depends directly upon the accuracy of the measur- 
ed receiver-performance curve in figure D-la and the accuracy of the 
combiner-performance curves in figure D-lb. 

D. 1. 1 Estimated Performance Characteristics of 
Radio Receivers-With Diversity 

The statistical characteristics of the "combined" receiver-output 

signal-to-noise ratio, S /N , depend upon both the receiver char- 
& oc' oc ^ ^ 

acteristics and the combiner characteristics. 

The curves in figure D-lb are essentially the combiner-perform- 
ance characteristics and apply to maximal-ratio type of combining; 
these curves were obtained from Brennan's work [19^8] . Curve A-A 
is the distribution of the signal at the combiner input, curves B-B and 
C-C, are the corresponding distributions at the combiner output, for 
dual and quadruple diversity, respectively. Curve A-A may be con- 
sidered as the distribution of the signal-to-noise ratio at the combiner 



-179- 
input, instead of the distribution of the signal at this point. Proper 

phasing of the combiner -input signals is required in order to obtain 

accurate measurements of the radio receiver output signal-to-noise 

ratio. 

The system design curves presented in this report were developed, 

for non, dual, and quadruple diversity, using the curves A-A, B-B, 

and C-C in figure D- lb, by the method outlined in Section D. 1 . 

D.2 Estimated Performance Characteristics of the Decoder Unit, 
for a Time-Varying Input Signal-to-Noise Ratio. 

The number of message errors, and hence the average message 
error rate occurring during a sampling period, depends upon the mes- 
sage-channel signal-to-noise ratio, S /N , at the decoder-unit in- 
5 5 oc' oc 

put. More in detail, we need to know the relationship between the 
average message-error rate and, 

(1) the value of the average signal-to-average noise power ratio, 

S /N , and 
oc oc 

(2) the statistical characteristics of: 

(a) the message-channel signal power, S , 

(b) the message-channel noise power, N , 

(c) the message-channel signal-to-noise ratio, S /N 

x ' 6 & oc' oc 

In this work it is assumed that the performance characteristic of 

the decoder unit for a time-varying (input) average signal-to-average 

noise power ratio, S /N , may be determined bv combining the de- 

oc oc 

coder-unit performance characteristic for the non-time -varying 

S /N condition, with the cumulative distribution of the time -varying 
oc oc 

S /N . It is assumed that the time interval required for an appreci- 
oc oc 

able change in S /N is large compared with the duration of a mes- 
6 oc' oc B * 

sage- signal binary pulse or the period of time required for one cycle 



-180- 
of the lowest frequency in the modulating message signal. The results 

obtained by the above procedure will also depend upon the statistical 

properties of the noise, N , at the decoder-unit input; each "type" of 

noise will have a different effect on the slope and the position of the 

decoder-unit performance-characteristic curve. 

Measurements of the average mess age -channel signal-power level, 

S , and the average message-channel noise-power level, N , may 
oc oc 

easily be made with the proper type of power-measuring meters having 
appropriate time - constants . However, if either or both of the average 
power levels, S or N , vary (independently) with time, their separ- 
ate measurements and also the measurement of the ratio, S /N , 

oc oc 

require special equipment capable of measuring the cumulative distri- 
bution of these variables. 

Following is an outline of the method used to calculate the perform- 
ance of a decoder unit for a time-varying signal-to-noise ratio, S /N , 

oc oc 

at the decoder-unit input. In this case, the decoder unit consisted of a 
teletype receiver and a teletype printer, combined. 

Figure D-2a shows the measured performance characteristic for 
the teletype-message decoder, for a range of values of steady signal- 
to-noise ratios, S /N , and assuming that N was thermal noise. 

oc oc oc 

Figure D-2b shows the estimated distribution of the time-varying 
signal-to-noise at the output of an FM radio receiver, for the non-di- 
versity case, and assuming a Rayleigh-fading type of signal at the 
radio receiver input. The method of obtaining this distribution, and its 

/ S oc\ 
average value, ( — — j , is shown in Section D. 1. Note that figure 

^ octave. 

D-2b is identical to figure D-lc, except that the ordinate scale values 

in figure D-2b are decreased over those in figure D- lc by the factor 

10 Log B (= 20.4 db). This scale modification is required so that 



-181- 



GRAPHICAL METHOD 
FOR DETERMINING MESSAGE- DECODER PERFORMANCt 



S3 * = 



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0.007 
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STEADY- SIGNAL MESSAGE-CHANNEL AVERAGE SIGNAL POWER 
MESSAGE-CHANNEL AVERAGE NOISE POWER 
MEASURED AT TELETYPE RECEIVER INPUT 
Fig. D-2o 



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00 I D 30 50 70 90 95 39 945 9995 99 

PERCENT OF TIME ORDINATE IS NOT EXCEEDED 

Fig. D-2c 



-182- 

the ordinate -scale values in figure D-2b correspond to the abscissa- 
scale values in figure D-2a. In other words, the procedures outlined 
in figure D-2 apply to a particular type of message-signal decoder 
unit. 

The distribution of the teletype-character error rate, shown in 
figure D-2c, was obtained as follows: In figure D-2b, an abscissa 
value of the percentage of time, t , was selected and the corresponding 

3, 

normalized value of (S /N ) was noted. The value of (S /N )', 

oc oc oc oc 

from figure D-2b, was used in figure D-2a to determine E'. This 
value of E' was then plotted versus t as shown in figure D-2c. The 

3. 

above procedure was repeated to obtain other points, E", E" 1 , etc. , 

for the curve shown in figure D-2c. 

The average value of the distribution shown in figure D-2c, E , 

B 6 ave. 

was obtained by numerical integration of the curve. This value of E 

y 6 ave. 

together with the value of (S /N ) . from figure D-2b, were plotted 
5 v oc' oc'ave B ^ 

as shown in figure D-2d. Additional points for the non-diversity curve 

in figure D-2d were obtained using the above procedures and the non- 

S 
oc 
diversity distribution in figure D-2b, but with different 



values for this distribution. Various values may be obtained for 

(S /N ) in figure D-2b by proper adjustment of the ordinate 

oc oc ave . 

scale values. 

Teletype decoder-unit performance curves for dual and quadruple 
diversity, shown in figure D-2d, were obtained using the same pro- 
cedure as outlined above and using S /N distributions, in figure D-2b, 

oc oc 

for dual and quadruple diversity, from figure D-lc. 



U.S. GOVERNMENT PRINTING OFFICE 19« — *47SM 



■ 



. S. DEPARTMENT OF COMMERCE 

Luther H. Hodges, Secretary 




NATIONAL BUREAU OF STANDARDS 

A. V. Astin, Director 

THE NATIONAL BUREAU OF STANDARDS 

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and 
Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work. 
In general, each section carries cut specialized research, development, and engineering in the field indicated by 
its title. A brief description of the activities, and of the resultant publications, appears on the inside of the 
front cover. 

WASHINGTON, D. C. 

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Dielectrics. High Voltage. 

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology. 
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Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment. 
Nucleonic Instrumentation. Neutron Physics. 

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ence Materials. Applied Analytical Research. Crystal Chemistry. 

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erations Research. 

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Engineering Applications. Systems Analysis. 

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Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- 
ments. Basic Instrumentation. 

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cular Kinetics. Mass Spectrometry. 
Office of Weights and Measures. 

BOULDER, COLO. 

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryo- 
genic Technical Services. 

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search. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical 
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Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval 

Standards. Millimeter-Wave Research. 

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration 

Center. 









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